"Any" is a somewhat ambiguous term that has caused a great deal of consternation. In general, it means "all" or "every," but it may be best to avoid using it. The problem is if one reads, for example, "Show that any even integer is a sum of two odd integers," one could reasonably respond, "Hot dog! I'll show that 8 is!" This interpretation of "any" gives the choice to the reader, whereas the intent in most mathematical writing is that "any" means "no matter which one you pick," aka "every."

• For any circle, the ratio of the circle's circumference to its diameter is \(\pi\). Better: For every circle, the ratio of the circle's circumference to its diameter is \(\pi\).
• Prove that any integer greater than 1 is either prime or composite. This does not mean "pick one and prove it works"! Better: Prove that every integer greater than 1 is either prime or composite.

In mathematics, "arbitrary" means "no matter which one is chosen" or "undetermined." That is, if a prompt refers to something arbitrary, it does not mean you can pick whichever one you want; it means that it has to be true for every possible choice or even without making a specific choice ("undetermined"), as though some adversary were choosing for you and trying to trip you up. You can often interpret "an arbitrary" as "every" or "all." Note that this is yet another term with a common meaning in English that can be deceptive in a mathematical context: in math, arbitrary does not mean it is up to your whim.

• (Algebra) Let \(f(x)=ax^2+bx+c\) be a quadratic function, where \(a, b\), and \(c\) are arbitrary constants. This means we won't assign \(a, b\), or \(c\) specific values but instead leave them as unknown constants.
• (Geometry) (Ceva's Theorem) Let \(\triangle ABC\) be an arbitrary triangle and \(D, E\), and \(F\) be points on the sides \(\overline{AB}, \overline{AC}\), and \(\overline{BC}\), respectively. Then \(\overline{AD}, \overline{BE}\), and \(\overline{CF}\) all meet at a single point \(P\) if and only if \(\frac{AF}{FB}\cdot\frac{BD}{DC}\cdot\frac{CE}{EA}=1\). (Note that here, a line over a pair of points indicates the line segment joining them; the absence of such a line indicates the distance between them.)

  

By inspection means that we don't need to employ any special techniques to solve something; we can find answers fairly quickly by a mental calculation or a guess-and-check method -- "just by looking." It's a little hazardous to use, like "obviously," in that "fairly quickly" is in the eye of the beholder. However, it's usually followed by an actual solution that can be checked.

• (Algebra) We can see by inspection that the solutions to \((x-1)(x-2)=0\) are \(x=1\) and \(x=2\). Since the equation is given to us in factored form, we can read off solutions just by knowing that one of the factors (either \(x-1\) or \(x-2\)) must be 0. Had it not been factored, we would have had to do some additional work, and the solution would not have been "by inspection."
• (Algebra) Consider the system of equations \[ \left\{ \begin{array}{rcc} x+y&=&2 \\ y&=&1\end{array}\right.. \] We find by inspection that \(y=1, x=1\) is a solution. We do have the small step of substituting \(y=1\) into the first equation, but then \(x+1=2\) is something we can likely solve in our heads.
• (Calculus) We could evaluate an integral like \(\int e^{2x} dx\) using \(u\)-substitution, but, knowing that exponential functions have exponential antiderivatives, we could also find by inspection that \(\int e^{2x} dx =\frac{1}{2}e^{2x}+C\).

Cancellation refers to algebraically removing something from an expression or equation. More formally, it means combining something with its inverse via some operation to result in the identity for that operation. Adding 3 to \(-3\) (its additive inverse) results in 0, the identity element for addition (adding 0 to any number does not change that number). Multiplying 5 by \(1/5\) (its multiplicative inverse) results in 1, the identity element for multiplication (multiplying any number by 1 does not change that number).

• We can cancel \(x\) from both sides of the equation \(3x^2+x=2x^2+x+1\) by adding \(-x\) to both sides of the equation: \(3x^2+x+(-x)=2x^2+x+1+(-x)\) simplifies to \(3x^2+0=2x^2+0+1\), or just \(3x^2=2x^2+1\). Most of the time, we don't bother with the step of adding \(-x\) to both sides and simplifying \(x+(-x)\) to 0; we just "cancel" the \(x\)s because we know how it will play out. Note that the intermediate step gets us a \(+0\) that we can then just ignore since 0 is the additive identity.
• We can cancel \(6\) from both sides of the equation \(6\sin(x)=6\) by multiplying both sides by \(\frac{1}{6}\): \(\frac{1}{6}(6\sin x)=\frac{1}{6}(6)\), which simplifies to \(1\cdot \sin x=1\), or just \(\sin x=1\). Most of the time, we don't bother with the step of multiplying both sides by \(\frac{1}{6}\) and simplifying \(\frac{1}{6}\cdot 6\) to 1; we just "cancel" the \(6\)s because we know how it will play out. Note that the intermediate step gets us a \(\cdot 1\) that we can then just ignore since 1 is the multiplicative identity.
• We can also cancel within an expression: \(\frac{x^2-3x+2}{x^2-4x+3}=\frac{(x-1)(x-2)}{(x-1)(x-3)}=\frac{x-1}{x-1}\cdot \frac{x-2}{x-3}=1\cdot \frac{x-2}{x-3}=\frac{x-2}{x-3}\). Usually, once the numerator and denominator are factored, we just cancel the common factor (\(x-1\) in this case) rather than separate the fractions, reduce to 1, and drop the 1. I included those steps in this example to illustrate that we are still taking advantage of multiplicative inverses (\((x-1)\) and \(\frac{1}{x-1}\) here) and the multiplicative identity to simplify the expression.
• A "famous" joke cancellation: \(\frac{\sin x}{n}=6\). This is, of course, not actually valid mathematically...

An expression is in closed form if it can be evaluated in a fixed (finite) number of steps. Things that we call "formulas" are generally in closed form. The basic idea is that if you can just "plug in" your numbers and get a result, the formula is in closed form.

• The expression \(1+2+3+\ldots+n\) is not in closed form because the number of steps gets bigger as \(n\) gets bigger, so it is not a fixed number of steps. The "\(\ldots\)" is a tell-tale that the expression is not in closed form. However, there is a closed-form formula for this expression: \(1+2+3+\ldots+n=\frac{n(n+1)}{2}\). The closed-form formula requires three steps: adding 1 to \(n\), multiplying that result by \(n\), and then dividing by 2. (For our purposes, we don't really care about the actual number of steps, only that it's fixed.)
• (Calculus) Newton's method from calculus gives an iterative method for finding roots of polynomials (among other things). You can repeat the method as many times as needed to reach the desired level of precision, but it won't usually give you an exact answer. However, this is not a closed-form formula. In some cases, though, there is a closed form for a root. The most familiar such closed form is probably the quadratic formula: \(\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).
• The famous Fibonacci sequence, \(1, 1, 2, 3, 5, 8, 13, \ldots\), is defined in terms of a recurrence relation, where each new term depends on prior terms. If \(F_n\) represents the \(n\)th Fibonacci number, then \(F_1=1, F_2=1\), and for \(n\ge 2\), \(F_n=F_{n-1}+F_{n-2}\). (E.g., \(F_4=F_3+F_2\).) This description is not in closed form since the number of steps to find \(F_n\) increases as \(n\) increases (since we would need to include the steps needed to find \(F_{n-1}\) and \(F_n\) when we count the steps needed to get \(F_{n+1}\)). However, there is also a closed form formula for \(F_n\): \[ F_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right). \] Try it and see!

A corollary is a theorem that can be proved directly from a prior theorem, usually without changing the hypotheses much if at all. In many cases, you may even see that the proof is omitted because there isn't much to add.

• (Algebra) Theorem: if \(a\ne 0\), then \(ax^2+bx+c\) has roots given by \(\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) (the famous quadratic formula). Corollary: If the discriminant \(b^2-4ac=0\), then \(ax^2+bx+c\) has exactly one root. Note that in stating the corollary, the hypotheses of the theorem were not restated, but they are implied. Also, while deriving the quadratic formula takes significant algebraic effort, if we already know it then there is only a small step to the corollary: if \(b^2-4ac=0\), the formula becomes \(\frac{-b}{2a}\) and we don't get the \(\pm\) that could give two different roots. (Bonus corollaries: if the discriminant is positive, there are exactly two real roots; if the discriminant is negative, there are no real roots.)
• (Geometry) Theorem: an inscribed angle in a circle has a measure equal to half of the central angle opening onto the same arc. In the figure on the left, central angle \(\angle CAB\) and inscribed angle \(\angle CDB\) both open onto ("subtend") minor arc \(\overparen{CB}\). Corollary: An inscribed angle is a right angle if and only if it opens onto a diameter. Again, while it takes some effort to prove the original theorem, once we know it we get the corollary nearly for free: a diameter is also a \(180^\circ\) central angle, so an inscribed angle opening onto it is \(90^\circ\) (and conversely). In the figure on the right, \(\angle D\) is a right angle since \(\overline{CB}\) is a diameter.
         
  
• (Calculus) Theorem: If \(f:\mathbb{R}\rightarrow\mathbb{R}\) is differentiable, then \(f\) is continuous. Corollary: Every polynomial is continuous. This kind of corollary narrows the focus of the theorem, in this case to a specific class of functions. Proving directly that an arbitrary polynomial is continuous in a calculus class would be pretty daunting, but proving they have derivatives isn't quite so difficult (or is at least approachable). Once that's done, the corollary gives us continuity without additional work.

A correlation between two measurable quantities is an indication of the strength of a (linear) relationship between them. In mathematics (statistics, really), it does not have the standard English meaning of "association" or "connection." A positive correlation indicates that the quantities grow or shrink together; a negative correlation indicates that as one grows, the other shrinks. The strength of the correlation is also measurable; take a course in statistics to delve into that.

• For any given person, there is a strong positive correlation between time spent reading a book and the number of pages read. Likewise, there is a strong negative correlation between time spent reading a book and the number of pages remaining.
• There is no correlation between how much spam I eat and how much spam email I get in a day.

To derive a formula is to deduce it from some starting point. Other results can also be derived; generally, there is a sense of discovery associated with derivations that is missing from proofs even though they may share some common elements.

• (Algebra) "Derive the quadratic formula" means to start with a general quadratic equation of the form \(ax^2+bx+c\) with \(a\ne 0\) and apply algebraic techniques to determine what the solutions have to look like. The derivation constitutes a proof that if \(x\) is a solution of that equation, then \(x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\), but the "derive" context implies the focus is on finding the formula rather than that proof.
• (Calculus) The act of finding a derivative of a function is differentiation, not "deriving." (Note that this time, "deriving" was not bolded because I don't want to encourage that error.) However, we can derive a formula for the derivative of, say, \(x^n\) (\(n\) a positive integer) by applying the definition of the derivative: \(\frac{d}{dx}(x^n)=\lim\limits_{h\rightarrow 0}\frac{(x+h)^n-x^n}{h}=\ldots=nx^{n-1}\). The "\(\ldots\)" represents the bulk of the work in the actual derivation, which I won't include here.

Epsilon is the fifth letter of the Greek alphabet; it is traditionally used in analysis to represent a tiny positive number. It has come to have a colloquial use in math to mean something really small.

• Paul Erdös famously referred to children as epsilons; there is an "Epsilon Camp" for children interested in math.
• "The probability that I care is less than epsilon."
• "I'm about epsilon away from finishing my homework."

In order for things to be equal, they must be the same: of the same kind and having the same value(s) in all respects. Equality is denoted with an equal sign, \(=\), which functions as a verb in a mathematical sentence. Caution: the equal sign is often abused by placing it between quantities that are not equal, as though it meant "Here's my next thought." It does not mean that.

• (Algebra) The polynomials \((x-1)(x+2)\) and \(x^2+x-2\) are equal by the Distributive Law ("FOIL"ing).
• (Algebra) The functions \(f(x)=\frac{x}{x}\) and \(g(x)=1\) are not equal even though \(f\) can be simplified to \(g\). The reason is that the domain of \(f\) is \((-\infty,0)\cup(0,\infty) \), while the domain of \(g\) is \((-\infty,\infty)\). Thus, they are not the same in all respects.
• (Linear Algebra) Two matrices are equal if they are the same size and their corresponding entries are all equal.
• Since a vector isn't a function, a function and a vector can't be equal. The same applies to a polynomial and a matrix or an ordered pair and a number.
• Here is an abuse of the equal sign. I'm hesitant even to include this lest someone think I'm endorsing it. To be clear: NEVER NEVER DO THIS! "Solving \(6x+1=13\): \(6x+1=13=6x=12=x=2\)." Just typing that gives me the willies. The second and fourth \(=\) should instead be "implies" or "gives" or "leads to" or "\(\implies\)," or even just starting a new sentence, but definitely not \(=\).

An equation is a mathematical sentence that expresses equality between two quantities. Equations involve an equal sign (\(=\)), which acts as the verb of the sentence; if there is no equal sign, it isn't an equation.

• (Algebra) The equation \(4x-5y=2\) is linear. The equation \(3x^2-7x+5=1\) is quadratic.
• (Algebra) The famous quadratic formula, \(\frac{-b\pm\sqrt{b^2-4ac}}{2a}\), is not an equation when expressed in that form (which is usually how it is memorized). But the solutions of the quadratic equation \(ax^2+bx+c=0\) are given by the equation \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) (when \(a\ne 0\)). Note the "\(x=\)" part; it provides the left-hand side of the equation.
• (Algebra) A system of equations is a collection of equations that are all supposed to have the same solution(s). For example, $$\begin{array}{ccc} 2x-5y&=&-1 \\ 4x+3y&=&11 \end{array}$$ is a system of equations with the solution \(x=2\) and \(y=1\). Those values satisfy both equations.
• (Calculus) A definite integral like \(\int_0^32xdx\) is not an equation, but \(\int_0^32xdx=9\) is. Note the presence of the equal sign.
• (Trigonometry) Functions can be defined in many ways, including via equations. For example, consider the function \(f\) defined by the equation \(f(x)=3\sin(2x-1)\).

We say that one number \(a\) divides another number \(b\) if there is a third number \(c\) such that \(b=ac\). Another way of saying that \(a\) divides \(b\) is to say that \(b\) is a multiple of \(a\). We could also say that \(a\) evenly divides \(b\); this means the same thing. It does not imply that the quotient (result) is itself an even number. Here, "evenly" is used to mean "with no remainder" or "with a remainder of 0."

• 6 divides 30 evenly since \(30=6\cdot 5\) (and there is no remainder). However, 6 does not divide 32 evenly since there would be a remainder of 2.
• This notion extends to other contexts, as well: \(x+2\) divides \(x^2+3x+2\) evenly since \(x^2+3x+2=(x+2)(x+1)\).

See verify.

"Exponentiate" is a term that pretty much does what it sounds like it should: to exponentiate an expression is to make it the exponent of some base, usually \(e\). It often arises in the context of exponentiating both sides of an equation to eliminate a logarithm.

• (Algebra) To solve \(\ln(3x)=-2\), exponentiate both sides: \(e^{\ln(2x)}=e^{-2}\). Since \(e^x\) and \(\ln x\) are inverses, we get \(2x=e^{-2}\) and thus \(x=\frac{e^{-2}}{2}\).

A mathematical expression is a computation of some kind, possibly implied, that can involve numbers, functions, operations, variables, etc. An expression differs from an equation in that it does not involve an equal sign; each side of an equation is an expression.

• (Algebra) Algebra is a common source of expressions. For example, \(3x^2-5x+2\), \(\sin(x)\), \(\frac{2x+1}{e^{4x}}\), and \(4\cdot 2+5\) are all expressions.
• (Calculus) \(\frac{d}{dx}(e^{x^2}\sin x)\) and \(\int_0^1t^3dt\) are both expressions.

(noun) A factor is a number (or other object) that is multiplied by another number (or object). Factors are also called multiplicands, but only when they're in trouble.

(verb) To factor an expression is to write it as a product of factors (the noun kind).

• In the expression \(3\cdot 7\), both 3 and 7 are factors. We would factor 21 by writing it as \(3\cdot 7\).
• (Algebra) The quadratic polynomial \(x^2-5x+6\) factors as \((x-2)(x-3)\), which is called its factored form. Its factors are \(x-2\) and \(x-3\).
• (Linear Algebra) LU decomposition is a way of factoring a matrix into a product of a Lower triangular matrix and an Upper triangular matrix (hence the LU). For example, \[ \left(\begin{array}{ccc}2&3&-1\\2&2&0\\4&2&6\end{array}\right)=\left(\begin{array}{ccc}1&0&0\\1&1&0\\2&4&1\end{array}\right)\left(\begin{array}{ccc}2&3&-1\\0&-1&1\\0&0&4\end{array}\right).\]

Mathematicians use fixed in the somewhat "classical" sense of unchanging or constant (as opposed to "repaired").

• "...produced by the moon and the fixed stars in their courses..." (Ptolemy's "Tetrabiblos," Book I p.5 as translated by James Wilson, 1828).
• (Algebra) Let \(a\) and \(b\) be fixed, and consider the line given by \(y=ax+b\).
• (Calculus) Fix \(c\) between 0 and 1 and consider the values of \(x\) such that \(\int_0^x\sin(t)dt=c\).

In mathematics, formal refers to something that follows a particular form (not something that wears a tux). It can also be used in the context of a "formal" proof (or definition), meaning one that carefully checks all the boxes of proof (or definition) as opposed to an "informal" proof (or definition) that gives a more intuitive explanation but omits some of the details.

• (Calculus) Leibniz notation for derivatives (e.g., \(\frac{dy}{dt}\)) looks like a fraction, but isn't. Nevertheless, under the right hypotheses we treat these formally as fractions: \(\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}\), where we follow the form as though the \(dx\)s "cancel."
• (Calculus/algebra) In calculus, we only assign meaning to a power series if it converges. However, we can also talk about formal power series: things that look like (have the form of) a power series, but where we don't worry about convergence.
• (Vectors) For three-dimensional vectors \( u= < u_1,u_2,u_3 >\) and \( v= < v_1,v_2,v_3 >\), we define their cross product \(u\times v\) as \(< u_2v_3-v_3u_2,-(u_1v_3-u_3v_1),u_1v_2-u_2v_1 >\). If we set \(\hat{i}=<1,0,0>, \hat{j}=<0,1,0>,\) and \(\hat{k}=<0,0,1>\), we can also compute the cross product formally as a determinant: $$\left|\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\ u_1&u_2&u_3\\v_1&v_2&v_3\end{array}\right|$$ even though the first row is made up of vectors, not numbers. If we follow the form of determinant computation, we come out with the right result.
• The sum of an even integer and an odd integer is odd since you can think of the even number as a bunch of pairs that look like \(1+1\) and the odd number as the same but with an extra \(+1\). In their sum, you just get lots of \(1+1\) pairs but still have the extra \(+1\), making the total odd. Formally, if \(a\) is even and \(b\) is odd, then there are integers \(m\) and \(n\) such that \(a=2m\) and \(b=2n+1\). Thus \(a+b=2m+(2n+1)=(2m+2n)+1=2(m+n)+1\), which is odd.

A formula is an equation (or expression) used to calculate something specific.

• The famous quadratic formula is used to calculate the solutions to a quadratic equation \(ax^2+bx+c=0\).
• We also have a formula giving the area of a circle of radius \(r\): \(A=\pi r^2\).
• The period \(T\) of a pendulum of length \(L\) is given by the formula \(T=2\pi \sqrt{\frac{L}{g}}\), where \(g\) is the acceleration due to gravity.

In mathematics, a function is a very specific way of pairing things up, which differs dramatically from its normal English usages (e.g., purpose (noun) or work correctly (verb). There are a lot of equivalent definitions people use. The most common formal definition is the "Bourbaki [Note: external link] definition: a function \(f\) from a set \(A\) to a set \(B\) is a set of ordered pairs \((x,y)\) with \(x\in A\) and \(x\in B\) such that every \(x\in A\) appears exactly once as the first element in the set of ordered pairs. We call \(A\) the domain and \(B\) the codomain, and we call \(x\) the input and \(y\) the output. Rather than write \((x,y)\in f\), we usually write \(f(x)=y\), pronounced "f of x equals y."

• There is no requirement that the sets \(A\) and \(B\) be sets of numbers. A function could pair each person in a class with whether or not they wear a watch. This might look like \(A=\{\)Colin, Josh, Inga, Kathryn, Peter, Erin, Heshan\(\}\) and \(B=\{\)yes,no\(\}\), and, say, \(f=\{\)(Colin,yes),(Josh,no),(Inga,no),(Kathryn,no),(Peter,yes),(Erin,no),(Heshan,yes)\( \} \).
• Functions are often described as "reliable machines" in the sense that a given input always produces the same output: if you give \(f\) the input 3 and it gives the output 7, then if you give \(f\) the input 3 tomorrow it will again give the output 7. No matter when or how many times you "plug in" 3, \(f\) will always return 7. (This of course assumes that \(f\) is fixed.)
• By viewing functions as ordered pairs, it becomes possible to graph them. If the domain and codomain are subsets of real numbers, the point \((x,f(x))\) lies in the Cartesian plane and can be graphed accordingly; the set of all \((x,y)\) in the plane such that \(y=f(x)\) is referred to as the graph of \(f\). This is often a way to help our brains understand the behavior of a complicated function. Traditionally, we associate the domain of the function with the horizontal axis and the codomain with the vertical axis. Note that we can graph any collection of ordered pairs (called a relation), but those graphs won't all represent functions. To determine whether a graph does represent a function, we have the Vertical Line Test, coming up next.
• The "Vertical Line Test" is just a graphical interpretation of the definition of a function. It says that no vertical line can intersect the graph of a function in more than one point. The reason for this is that all points on a vertical line have the same \(x\)-coordinate, so a vertical line hitting a graph twice means that graph cannot be the graph of a function since the same domain element would correspond to two different codomain elements.
• Even more formal: a function \(f\) from a set \(A\) to a set \(B\) is a subset \(f\) of \(A\times B\) such that if \(x\in A\) then there is one and only one \(y\in B\) such that \((x,y)\in f\).

"If and only if" is a term of logic that says two statements are logically equivalent -- saying one is like saying the other, and vice versa. If we label the statements \(p\) and \(q\), then "\(p\) if and only if \(q\)" means that if \(p\) is true, so is \(q\), and if \(q\) is true, so is \(p\). This is not the case in general for "if-then" statements.

• (Geometry) Pythagorean Theorem: Let \(a, b\), and \(c\) be the side lengths of a triangle. Then the triangle is a right triangle with hypotenuse length \(c\) if and only if \(a^2+b^2=c^2\). This says that if the triangle is a right triangle with hypotenuse \(c\), then \(a^2+b^2=c^2\), and it also says, conversely, that if \(a^2+b^2=c^2\), then the triangle is a right triangle with hypotenuse \(c\).
• (Algebra) For fixed \(a\) and \(b\) with \(a\ne 0\), \(ax+b=0\) if and only if \(x=-\frac{b}{a}\).
• (Trigonometry) If \(x=\pi/2\), then \(\sin(x)=1\). However, it is not the case that if \(\sin(x)=1\), then \(x=\pi/2\) since \(\sin(5\pi/2)=1\), as well. Thus, this is not an "if and only if" scenario.

We say in general to mean "other than in special cases" (or "in cases besides the one mentioned") although it can sometimes mean "always." In the former usage, we usually mean that the special cases are rare.

• Although 2 is prime, in general, even numbers are not prime.
• (Geometry) Parallel lines do not intersect (unless they are actually the same line), but, in general, two lines intersect in exactly one point.
• (Algebra) A system of two linear equations in two unknowns has no solutions (or infinitely many) if the lines have the same slope, but, in general, such a system has exactly one solution. (Note that this is just the previous example recast in algebraic terms.)
• (Calculus) Every differentiable function is continuous, but continuous functions are not in general differentiable.
• (Geometry) In general, three points in the plane determine a triangle unless they happen to be collinear.

"Express Y in terms of X" means "Write Y in a way that shows how Y depends on X." It often appears in the form, "Solve for \(y\) in terms of \(x\)," which usually just means "solve for \(y\)" since the "in terms of \(x\)" part will happen automatically. However, if there are more variables or other expressions involved, "in terms of \(x\)" may be used to clarify that that particular dependence is the one we're most interested in. It may also be that we want to express Y in terms of multiple things.

Sometimes, we mean that we want the solution to have a particular form; see the second example below.

Finally, sometimes there is no equation to solve. Rather, we are attempting to establish a relationship between Y and X, so we are creating an equation. If the variables are \(y\) and \(x\), we have an expectation (not always correct) that we want to express \(y\) in terms of \(x\). When the variables are something else, we have no expectation to fall back on, so we need to know which variable to express in terms of the other(s).

• Given \(PV=nRT\), express \(V\) in terms of \(P, n, R\), and \(T\). We would get \(V=\frac{nRT}{P}\). Alternatively, we could have been asked simply to express \(V\) in terms of \(P\), and we would have the same result, with \(n, R\), and \(T\) along for the ride.
• (Algebra) Express the polynomial \(x^6-3x^4+5x^2-1\) in terms of \(x^2\). We would rewrite the polynomial as \((x^2)^3-3(x^2)^2+5x^2-1\), so each \(x\) actually appears as an \(x^2\) but the polynomial remains the same.
• (Algebra) Solve for \(y\) and express your answer in terms of \(e^x\): \(\ln(xy)=x\). Here, we would exponentiate both sides to get \(xy=e^x\), giving \(y=\frac{e^x}{x}\). However, note that the denominator involves \(x\) but not \(e^x\). We need to rewrite it as \(x=\ln(e^x)\) to get \(y=\frac{e^x}{\ln(e^x)}.\) Now, everywhere that \(x\) appears, it appears in the desired form \(e^x\), so our answer is in terms of \(e^x\).
• On an average day, 600 cars visit Quick Coffee, and they spend an average of $4.50. Express the total revenue \(R\) taken in by Quick Coffee in terms of the number \(d\) of days that have gone by.

  Here, we must create an equation in which \(R\) depends on \(d\). We would have \(R=600\cdot 4.5d=2700d\).

Integral has two primary meanings in mathematics. There is the calculus meaning, referring to a definite or indefinite integral (\(\int_a^bf(x)dx\) or \(\int f(x)dx\)). Integral is also used as the adjective form of integer. This second meaning is the focus of this entry; think of integral in this sense as meaning "whole."

• The set of integral solutions to \(3x+4y=12\) is different from the set of real-number solutions. Note that you may also see this written as "The set of integer solutions to..." They mean the same thing.
• The "floor" function, also known as the "round-down" function, returns the integral part of its input. For example, the floor of 2.75 is 2, the floor of \(-2.75\) is \(-3\), the floor of 6 is 6, and the floor of \(\pi\) is 3.

See verify.

In an expression, like terms are terms that are the same except possibly for a constant multiplier. They can be simplified using the Distributive Law, giving rise to the prompt, "Combine like terms."

• \(4x^2\) and \(7x^2\) are like since both are \(x^2\) with a constant multiplier. If they are being added, then they are like terms and can be combined: \(4x^2+7x^2=(4+7)x^2=11x^2\).
• In the expression \(6\sin(3x)-8\cos(x)+2\sin(x)-4\sin(3x)\), \(6\sin(3x)\) and \(-4\sin(3x)\) are like terms since both are \(\sin(3x)\) with a constant multiplier. The \(\sin(x)\) is not like these since it is not a constant multiple of \(\sin(3x)\).
• Since \(\sqrt{4x}=2\sqrt{x}\), \(\sqrt{4x}\) and \(5\sqrt{x}\) are like terms even though they initially don't look like it; it just takes a little algebraic manipulation first. We could combine them: \(\sqrt{4x}+5\sqrt{x}=2\sqrt{x}+5\sqrt{x}=(5+2)\sqrt{x}=7\sqrt{x}\).

Linear is used in a few different ways. Algebraically, it refers to first-degree equations or expressions. Geometrically, it refers to things that are "like" lines; i.e., things that are straight or flat. In terms of functions, linear means a function that grows at a constant rate (i.e., increasing the input by a 1 results in a consistent increase in the output regardless of the input's starting point). There is also a more specialized meaning in Linear (there's that word again!) Algebra for "linear transformations"; see the Linear Algebra example below. Linearity refers to the algebraic property that such functions have; the last example discusses this.

• (Algebra) The equation \(4x-2y=5\) is linear since \(x\) and \(y\) only appear to the first power (and are not multiplied together). Likewise, \(4w+3x=y-3z+2\) is linear since all variables appear to the first power only (and are not multiplied together).
• (Algebra) However, \(3xy=1\) is not linear even though they both appear to the first power. The reason is that the degree of a term is its total degree -- the sum of all degrees appearing in that term. The degree of \(3xy\) is 2 because the (invisible, but present) exponent on \(x\) is 1, as is the exponent on \(y\), and \(1+1=2\). An equation like \(\sin(x)=e^y\) is also not linear because the variables appear inside of functions that are themselves not linear -- their graphs are not lines -- even though both \(x\) and \(y\) "look like" they are to the first power. Similarly, \(\frac{1}{x}\) is not linear even though it "looks" like \(x\) appears only to the first power since in fact \(\frac{1}{x}=x^{-1}\).
• (Geometry) The graph below on the left is linear, while the one on the right is not.
         
• (Geometry) The equation \(4x-2y+3z=3\) is linear since each variable appears only to the first power (and none are multiplied together). The graph of the solution set of this equation is a plane in space (see below), which is flat.
• (Algebra) The function \(f(x,y)= 12x-5y+3\) is linear since both \(x\) and \(y\) appear only to the first power (and are not multiplied). Notice that changing \(x\) but not \(y\) gives \(f\) a constant rate of change, as does changing \(y\) but not \(x\); e.g., \(f(2,1)=22, f(3,1)=34, f(4,1)=46\), etc., an increase of \(f\) by 12 for each increase of \(x\) by 1. Similarly, \(f(2,1)=22, f(2,2)=17, f(2,3)=12\), etc., a decrease of \(f\) by 5 for each increase of \(y\) by 1.
• (Algebra) The function \(f(x,y)=x^2+6y-2\) is not linear since the \(x\) is squared, but we can say that it is linear in \(y\). This phrasing acknowledges that if we increase \(x\) (and hold \(y\) constant), the function does not grow at a constant rate, while if we increase \(y\) (and hold \(x\) constant), it does. This allows us to specify linear behavior within a nonlinear setting.
• (Linear Algebra) A linear transformation is a function \(T\) between vector spaces with the property that \(T(x+y)=T(x)+T(y)\) and \(T(cx)=cT(x)\) for all \(x,y\) in the domain and all scalars \(c\). This is a little more specialized. For example, if we take both vector spaces to be \(\mathbb{R}\) with real number scalars, then \(T\) given by \(T(x)=3x\) is a linear transformation
  (\(T(x+y)=3(x+y)=3x+3y=T(x)+T(y)\) and \(T(cx)=3(cx)=c(3x)=cT(x)\) for all scalars \(c\) and vectors \(< x,y >\)), but \(T\) given by \(T(x)=3x+1\) is not (\(T(x+y)=3(x+y)+1\), while \(T(x)+T(y)=(3x+1)+(3y+1)=3(x+y)+2\)). This may feel a bit peculiar: \(T:\mathbb{R}\rightarrow \mathbb{R}\) is a linear function, but it is not a linear transformation. If you're studying linear algebra, you'll need to be aware of that distinction to avoid falling into the trap of thinking linear functions are also linear transformations -- they are not necessarily.
• (Calculus) Derivatives and integrals are linear in the sense of linear transformations above: \((f(x)+g(x))'=f'(x)+g'(x)\) and \(\int (f(x)+g(x))dx=\int f(x)dx+\int g(x)dx\), and \((cf(x))'=cf'(x)\) and \(\int cf(x)dx=c\int f(x)dx\). Trig functions and square roots, on the other hand, are not linear: \(\sin(x+y)\) and \(\sin(x)+\sin(y)\) are not the same, and \(\sqrt{x+y}\) and \(\sqrt{x}+\sqrt{y}\) are not the same.

A "map" usually just means a function, but map can also have more technical meanings in different areas of mathematics. (In the more technical cases, the precise definition should be spelled out for you.) The difference is that "map" tends to evoke a more dynamic image of what is happening: thinking of individual elements being "mapped" to other elements gives a different perspective from the usual input/output conception of a function. We will usually use "map" when we want to describe what happens to individual elements. Unlike the mathematical version of "function," we can also use "map" as a verb, and that is probably more common.

• (Vector analysis) The function \(r\) given by \(r(t)=<\cos(t),\sin(t),t>\) maps the real number \(t\) to a vector in space.
• (Geometry) A dilation of the plane maps any given figure to a new figure that is similar to the first, and the constant of proportionality is the scale factor of the dilation.

A mathematical model is an equation, expression, function, graph, or any mathematical structure (or a system of them) that purports to describe some phenomenon. It usually involves some simplifying assumptions since the world is super complicated, and models usually have various limitations.

• In physics, we model the force \(F\) exerted on an object by a spring displaced \(x\) units from equilibrium by \(F=kx\), where \(k\) is a constant that depends on the stiffness of the spring. This model includes some limitations: it doesn't make sense if \(x\) is a billion miles (assuming we're using a spring we can actually manufacture), but, perhaps more subtly, it isn't accurate if the spring is stretched or compressed too far. Nevertheless, for small displacements, the model gives good agreement with measurements.
• One model for the density \(d\) of the atmosphere \(r\) units from the center of Earth looks like \(d=Ae^{-kr}\), where \(A\) and \(k\) are constants that depend on the choice of units. This model will give good agreement with measurements of atmospheric density most of the time, but doesn't account for weather variations. It also gives an atmospheric density 2 miles underground, which makes no sense, and implies Earth's atmosphere extends outward forever.
• We can model the relationships among a collection of chemicals with a graph in which the vertices (points) represent the chemicals and the edges join vertices that are dangerous to store together:

When we ask for something "of the form" (or "in the form") of something else, we mean that we want the result expressed in a way that is structurally similar.

• (Algebra) Find a positive solution to \(x^2-2x-1=0\) of the form \(a+b\sqrt{c}\). The problem specifies the structure of the solution, so, while a decimal answer could be (at least approximately) a correct solution, it would not be of the required form. In this case, the quadratic formula gives \(x=\frac{2\pm\sqrt{8}}{2}=1+1\sqrt{2}\). We can see that \(a=1, b=1, c=2\).
• (Algebra) What is the slope of the line through \((-1,2)\) and \((4,5)\)? Express your answer in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers. Here again, note that while a decimal answer could be correct, the problem explicitly asks for an answer that has the structure of a fraction. The slope is \(\frac{5-2}{4-(-1)}=\frac{3}{5}\).

A product is the result of multiplying things together. (Contrast with a sum, which is the result of adding things together.) The things being multiplied are the factors.

• The product of 5 and 7 is 35.
• (Algebra) The product of \(x-1\) and \(2x+3\) is \((x-1)(2x+3)\). It is also \(2x^2+x-3\); both forms represent the product.

One quantity is proportional to another (or the two are proportional) if it is a constant multiple of the other. In symbols, \(a\) is proportional to \(b\) if there is a constant \(k\), called the constant of proportionality, such that \(a=kb\). In this situation, if \(b\) doubles, so does \(a\); if \(b\) triples, so does \(a\), etc. But if \(b\) increases by 2, \(a\) increases by \(2k\): \(k(b+2)=kb+2k\). Any percentage change in \(b\) is matched by the same percentage change in \(a\), but other changes in \(b\) have different effects on \(a\). We may also say that two quantities are in proportion if one is proportional to the other. Note that this also means that the ratio of the two quantities is constant: \(\frac{a}{b}=k\); this is another way of saying that \(a\) and \(b\) are proportional.

• (Physics) The force \(F\) exerted by a spring is proportional to its displacement \(x\) from equilibrium: \(F=kx\). Here, the constant of proportionality is called the spring constant.
• (Geometry) The fact that the side lengths of similar triangles are proportional allows us to define the slope of a line. In the figure, triangles \(ADE, AFG\), and \(ABC\) are all right triangles with a common angle at \(A\), so they are similar by the AA criterion and their sides are in proportion. That means that the ratio of, say, the vertical sides to the horizontal sides are all the same: \(\frac{ED}{AD}=\frac{GF}{AF}=\frac{CB}{AB}\). If we call that common ratio \(k\), we see that \(ED=k(AD), GF=k(AF)\), and \(CB=k(AB)\); in general, (vertical side)=\(k\)(horizontal side) -- the "rise" is proportional to the "run."
• (Trigonometry) In fact, that same proportionality between sides of similar triangles allows us to define all of the trigonometric ratios as functions since the angle \(\theta\) at \(A\) in the figure above determines the ratio no matter how large or small the triangles are. In particular, \(\tan(\theta)\) is the slope, \(\sin(\theta)=\frac{DE}{AE}=\frac{FG}{AG}=\frac{BC}{AC}\) and \(\cos(\theta)=\frac{AD}{AE}=\frac{AF}{AG}=\frac{AB}{AC}\).
• We can modify "proportional," too. For example, we may say that \(y\) is inversely proportional to \(x\), by which we mean that \(y\) is proportional to the (multiplicative) inverse of \(x\): \(y=k\cdot\frac{1}{x}=\frac{k}{x}\).

See verify.

In mathematics, "rational" refers to whether a number can be expressed as a ratio of integers. (More generally, it refers to whether something can be expressed as a ratio of integer-"like" things.) When a number cannot be so expressed, we call it irrational.

• The numbers \(\frac{3}{5}\) and \(0.7\) are rational since both can be expressed as a ratio of integers: \(3\) and \(5\) in the first case, and \(7\) and \(10\) in the second (since \(0.7=\frac{7}{10}\)).
• Numbers with repeating decimal expansions are rational since they can be written as a ratio of integers, like \(0.\overline{27}=\frac{27}{99}\).
• On the other hand, numbers with non-repeating decimals are irrational. This includes numbers like \(e, \pi, \sqrt{2}\) and \(0.101001000100001...\). The last one has a pattern in its decimal expansion, but not a repeating pattern.
• (Algebra) A rational function is one that can be expressed as a ratio of polynomials, like \(\frac{3x^2-7x+2}{4x^3-x-9}\). In the world of functions, polynomials are "integer-like" in ways that we won't go into here.

"Respectively" means "in the same order." It is used when we have a list of objects corresponding to a list of properties and we want to indicate that the first property goes with the first object, the second with the second, and so on. It is used this way outside of mathematics, as well.

• (Algebra) In the figure below, the functions \(f\) and \(g\) are odd and even, respectively. This means that \(f\) is odd and \(g\) is even.

  

• (Algebra) Let \(f, g\), and \(h\) be polynomials of degrees 3, 4, and 5, respectively. This means \(f\) has degree 3, \(g\) has degree 4, and \(h\) has degree 5.
• (Number Theory) On a number line, assign the equivalence classes of 0, 1, and 2 modulo 3 the colors red, green, and blue, respectively. This means the numbers congruent to 0 mod 3 will all be red, the numbers congruent to 1 mod 3 will all be green, and the numbers congruent to 2 mod 3 will all be blue.

"Satisfy" appears in many contexts, but the general idea is X satisfies Y if X meets the conditions described by Y.

• (Algebra) We say that 2 satisfies the equation \(4x-7=1\) since \(4\cdot 2-7=1\), meaning that 2 solves the equation, i.e., 2 meets the condition on \(x\) described by the equation \(4x-7=1\).
• Likewise, the system of equations \(3x-2y=2, 2x+y=13\) is satisfied by \(x=4, y=5\).
• Consider the following little theorem: if \(n\) is even, then \(n\) is the sum of 2 odd numbers. The number 12 is even, so 12 satisfies the hypothesis (the "if" part) of the theorem.

See verify.

"Simplify" is a slippery notion. The big picture is to take something given in a complicated form and turn it into something equivalent, but in a not-complicated form. The problem is that "not-complicated" depends somewhat on the context. For certain purposes, it may be "simpler" to leave an expression "unsimplified" because simplifying obscures useful information. In general, though, simplify means to reduce the number of terms or factors --usually as much as possible -- via cancellation so that the result is visually easier for our brains to digest and there are no redundant terms or factors. A simplified expression will generally have only one term of a given kind, and each term will have only one factor of a given kind.

• Simplifying \(6x-2x\) gives \(4x\).
• Simplifying \(\frac{6x^3y^2}{4xy^3}\) gives \(\frac{3x^2}{2y}\).
• Traditionally, "simplified" expressions don't have irrational expressions in the denominator; e.g., \(\frac{5}{\sqrt{2}}\) is not simplified, but it can be simplified by rationalizing the denominator: \(\frac{5}{\sqrt{2}}=\frac{5}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{5\sqrt{2}}{2}\). The reason for this is now archaic: dividing by \(\sqrt{2}\) by hand is not easy, but dividing by 2 is. However, with the advent of handheld calculators, this kind of "simplifying" is somewhat obsolete (in this person's opinion). Indeed, in a calculus class you may have occasion to rationalize the numerator instead!

We use such that to indicate the conditions or assumptions we are working under; when we say "X such that Y," we mean that the "X" we are talking about has the properties described by "Y."

• "Let \(f(x)=mx+b\) be a linear function such that \(m\ge 1\)" tells us that the linear function we are thinking about has a slope of at least 1.
• (Calculus) The Squeeze Theorem: Let \(f, g\), and \(h\) be continuous functions on the interval \((a,b)\) such that \(f(x)\le g(x)\le h(x)\) for all \(x\in (a,b)\). If \(c\in (a,b)\) and \(\displaystyle \lim_{x\rightarrow c}f(x)=\lim_{x\rightarrow c}h(x)=L\), then \(\displaystyle\lim_{x\rightarrow c}g(x)=L\), as well. (Note: we can actually exclude the point \(x=c\) from the hypotheses and still have the conclusion hold.)
• (Geometry) Let \(ABCD\) be a convex quadrilateral such that opposite sides are congruent. Then \(ABCD\) is a parallelogram.

A sum is the result of adding things together.

• The sum of 3 and 5 is 8.
• The sum of \(2x^2\), \(3x\), and \(5\) is \(2x^2+3x+5\).
• The sum of the integers from 1 through 20 is \(\displaystyle \sum_{n=1}^{20}n=\frac{20(21)}{2}=210\).

A term is one member of a sum -- one of the things being added together.

• The terms of the sum \(1+3+5\) are 1, 3, and 5.
• The terms of \(3x^2-\frac{7}{2}x+5\) are \(3x^2, -\frac{7}{2}x,\) and \(5\). Note that the term \(-\frac{7}{2}x\) includes the \(-\) sign.
• The terms of \(\sin(x)-3\sin(2x)+4e^{x^2}\ln(x)\) are \(\sin(x), -3\sin(2x),\) and \(4e^{x^2}\ln(x)\).

A "bound" refers to an upper or lower limit on the numerical value of some quantity. When we say that a bound is tight, we mean that there is no "better" bound possible. Contrast this with a sharp bound, which means that while a quantity can't go past a certain number, it does reach it. A tight bound is not necessarily reached.

• (Algebra) The number of real roots of a polynomial of degree \(n\) with real coefficients is bounded by \(n\). This bound is tight (and sharp) since for any nonnegative integer \(n\), the polynomial \((x-1)(x-2)\cdots (x-n)\) is a polynomial of degree \(n\) having exactly \(n\) roots.
• (Trigonometry) The inverse tangent function is bounded above by \(\pi/2\) and below by \(-\pi/2\). These bounds are tight since the arctangent approaches both of them asymptotically. However, neither is sharp since there is no value of \(x\) such that \(\arctan(x)\) is equal to \(\pm \pi/2\).
• (Graph Theory) If a connected graph with \(n\ge 3\) vertices can be drawn in the plane without having any crossing edges, then it has at most \(3n-6\) edges. This bound is tight (and sharp) since for any \(n\) such a graph exists with \(3n-6\) edges.

The term "uniformly" shows up in a few different contexts; this glossary is focused on its use in probability. When we say that numbers are chosen uniformly, we mean they are chosen with equal probability: any one of the numbers is as likely to be chosen as any other.

• What is the expected value of the average of four numbers chosen uniformly from the interval \([0,1]\)? Note that this does not mean the four numbers must be evenly distributed across \([0,1]\), but rather that they are chosen with equal probability. That is, it's the probability that's "uniform" rather than the distribution of the four numbers within the interval.

"Up to X" means "if we ignore X." It's often used to clarify what we mean when we claim there is only one of something, as in "unique up to X."

• We can factor any whole number greater than 1 into a product of prime numbers, and the product is unique up to order. This means that there is only one way to express the number as a product of primes, assuming we ignore the order in which we write the primes. In this situation, we would consider \(3\cdot 2\) the "same" as \(2\cdot 3\).
• (Calculus) In calculus, we could say that the antiderivative of a continuous function is unique up to an additive constant, meaning that there is only one antiderivative for a continuous function if we ignore any constant terms. In this situation, we would consider \(\sin(x)\) and \(\sin(x)+4\) as the "same" antiderivative of \(\cos(x)\). (In calculus, we would usually write \(\sin(x)+C\) to cover all of the different possible antiderivatives of \(\cos(x)\) at once.)
• (Number Theory) (Sun Zi's Theorem) If \(p\) and \(q\) are distinct prime numbers and \(a\) and \(b\) are integers, then there is a number \(x\) such that \(x\equiv a\pmod{p}\) and \(x\equiv b\pmod{q}\). Furthermore, \(x\) is unique up to congruence modulo \(pq\).

  For instance, if \(p=5\), \(q=11\), \(a=4\), and \(b=7\), then we may take \(x\) to be 29 since \(29\equiv 4\pmod{5}\) and \(29\equiv 7\pmod{11}\). But the solution is certainly not unique since it is also true that \(84\equiv 4\pmod{5}\) and \(84\equiv 7\pmod{11}\). However, note that \(84\equiv 29\pmod{5\cdot 11}\), which is all Sun Zi's Theorem claims: any solution other than 29 must be congruent to 29 modulo 55.

In mathematics, variance refers to a statistical measure of how spread out a data set is. However, there is a second meaning that shows up sometimes in the context of a technique known as "variation of parameters." Some authors will use "variance" to describe the process one uses in applying this technique. This is a technical area far beyond the scope of this glossary, so this entry is here for disambiguation in case someone gets confused by the secondary meaning.

Check with your teacher for the meaning of these since there is some variation. Generally, prove means "give a formal proof"; verify means to check that something works as claimed, which could involve a proof or just be a quick "Yeah, that makes sense"; and show means "give a (perhaps informal) proof." Explain can take a different direction, where the intention is to add discussion that can clarify technical details. Justify is somewhere in the middle ground. The ambiguity here is in the level of formality desired, which is why you need to check with your teacher.

• Prove that the sum of two arbitrary even integers is even. A proof might look something like this: Let \(a\) and \(b\) be two given even integers. Then there are integers \(m\) and \(n\) such that \(a=2m\) and \(b=2n\) by the definition of even. Thus the sum of \(a\) and \(b\) is \(a+b=2m+2n=2(m+n)\) by substitution followed by the distributive law. By the definition of even, \(2(m+n)\) is even, so the sum of \(a\) and \(b\) is even. Since \(a\) and \(b\) were arbitrary, the sum of two arbitrary even integers is even. This is quite formal; contrast with the next example.
• Verify that the sum of two arbitrary even integers is even. Since both even integers have a factor of 2, so does their sum. Whether I would consider that a proof would likely depend on the level of the class. In an advanced class, I would call it a proof; in an introduction-to-proof course, I would likely want more detail since I would not yet know how well the student had internalized the idea of proof. This example could also fall into the "explain" or "justify" category. So, once again, check with your teacher!
• Show that the angle sum in a (Euclidean) triangle is 180\(^\circ\). One common approach to this is to refer to the diagram below, in which the two horizontal lines are parallel and congruent angles are marked. The angles at the top have measures summing to 180 since they lie along a straight line, but they also match the angles within the triangle. I've omitted quite a few details; some teachers would consider this "shown" but not "proved" (due to the omitted details) and some would consider it "proved."

"With respect to \(x\)" means that \(x\) is the variable we are focusing on. There may or may not be other variables involved, but \(x\) is the one we are working with -- the one we are treating as "the" independent variable. This will usually follow a verb, e.g., "differentiate" ("take the derivative of") or "integrate." More generally, "with respect to" means "regarding"; it indicates where to focus your attention.

• Consider \(f(x)=4x^6\). Since there is only one variable, \(x\), it is unambiguous to say "Differentiate \(f(x)\),'' but it would be equivalent to say "Differentiate \(f(x)\) with respect to \(x\)." That just means that \(x\) is the independent variable here and the variable of differentiation.
• If, on the other hand, \(f\) is a function of two variables, "Differentiate \(f\)" is ambiguous. Take \(f(x,y)=x^3y^2\), for example. Here, we can differentiate with respect to \(x\) or with respect to \(y\). If we differentiate with respect to \(x\), we are treating \(x\) as though it were the only variable and \(y\) as though it were constant, so we would get \(\frac{\partial f}{\partial x}=3x^2y^2\). The \(\partial x\) notation tells us with respect to which variable we are differentiating.
• Likewise, \(\int x^3y^2\) would be ambiguous; we need to know with respect to which variable we are integrating. The differential gives us this information: \(\int x^3y^2 dy=\frac{1}{3}x^3y^3+C\). In this case, the \(dy\) tells us to treat \(y\) as the variable and \(x\) as though it were constant; we are integrating with respect to \(y\).
• In a closed system, energy is constant with respect to time. It is not, however, constant with respect to position.