Multivariable and Vector Calculus reference sheet

This sheet follows the six units of the course in order. Each unit starts with the big idea in a few sentences, then gives the definitions, formulas and theorems you will look up most often, with the conditions they need. When you want the full explanation, a proof or more examples, follow the lesson links under each topic.

Coordinates in space are right-handed and angles are in radians. Spherical coordinates are written $(\rho, \theta, \phi)$, with the azimuth $\theta$ measured in the $xy$-plane from the positive $x$-axis and the polar angle $\phi$ measured from the positive $z$-axis. A formula holds wherever both sides are defined unless a condition says otherwise.

Vectors and the Geometry of Space

Multivariable calculus happens in space, so it starts with a precise way to describe points, directions and flat or curved surfaces. This unit sets up coordinates and vectors, the dot and cross products that measure angles, areas and volumes, the matrices that describe linear maps, and the equations of lines, planes, quadric surfaces and round coordinate systems.

Points and vectors

The axes are right-handed. Curling the fingers of the right hand from the positive $x$-axis toward the positive $y$-axis leaves the thumb along the positive $z$-axis. With $\Delta x = x_2 - x_1$, $\Delta y = y_2 - y_1$ and $\Delta z = z_2 - z_1$, the distance between $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is

and the sphere with center $(h, k, l)$ and radius $r > 0$ is

An equation $x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0$ becomes this form after completing the square in each variable, and it is a sphere, a single point or empty as the right side comes out positive, zero or negative. For example $x^2 + y^2 + z^2 - 4x + 6y - 2z = 2$ is the sphere with center $(2, -3, 1)$ and radius $4$.

The vector from $A(x_1, y_1, z_1)$ to $B(x_2, y_2, z_2)$ is $\overrightarrow{AB} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle$, and a vector $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ equals $a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k}$ in the standard basis.

Forces add as vectors, and an object is in equilibrium when the resultant of all forces on it is $\mathbf{0}$.

Lessons Three-dimensional coordinates and vectors

The dot product

For $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$, and with $\theta$ in $[0, \pi]$ the angle between them when both are nonzero,

The product is commutative and distributive, and $\mathbf{a} \cdot \mathbf{a} = \lvert \mathbf{a} \rvert^2$.

The remainder $\mathbf{b} - \operatorname{proj}_{\mathbf{a}} \mathbf{b}$ is orthogonal to $\mathbf{a}$. Projecting $\mathbf{b} = \langle 1, 1, 2 \rangle$ onto $\mathbf{a} = \langle 1, 0, 1 \rangle$ gives $\left\langle \frac{3}{2}, 0, \frac{3}{2} \right\rangle$, and the remainder has length $\frac{\sqrt{6}}{2}$, the distance from $(1, 1, 2)$ to the line through the origin along $\mathbf{a}$.

The Cauchy-Schwarz inequality $\lvert \mathbf{a} \cdot \mathbf{b} \rvert \le \lvert \mathbf{a} \rvert \, \lvert \mathbf{b} \rvert$ holds for all vectors, with equality exactly when one vector is a scalar multiple of the other. It implies the triangle inequality $\lvert \mathbf{a} + \mathbf{b} \rvert \le \lvert \mathbf{a} \rvert + \lvert \mathbf{b} \rvert$.

Lessons The dot product

The cross product

For vectors in space, the cross product is the vector with the components listed below as a column,

It is perpendicular to both factors, $\mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b})$, it is distributive, and it is not associative. In right-handed coordinates $\mathbf{i} \times \mathbf{j} = \mathbf{k}$, $\mathbf{j} \times \mathbf{k} = \mathbf{i}$ and $\mathbf{k} \times \mathbf{i} = \mathbf{j}$. For nonzero $\mathbf{a}$ and $\mathbf{b}$ with angle $\theta$,

and when they are not parallel the product points where the right thumb points as the fingers curl from $\mathbf{a}$ toward $\mathbf{b}$. Two nonzero vectors are parallel exactly when their cross product is $\mathbf{0}$.

The triple product is zero exactly when the three vectors are coplanar, positive when $(\mathbf{a}, \mathbf{b}, \mathbf{c})$ is a right-handed triple and negative when it is left-handed. For the triangle with vertices $P(1, 0, 0)$, $Q(0, 2, 0)$ and $R(0, 0, 3)$, the product $\overrightarrow{PQ} \times \overrightarrow{PR} = \langle 6, 3, 2 \rangle$ has length $7$, so the area is $\frac{7}{2}$ and the plane through the three points is $6x + 3y + 2z = 6$.

Lessons The cross product

Matrices, linear maps and determinants

An $m \times n$ matrix $A$ with columns $\mathbf{a}_1, \dots, \mathbf{a}_n$ acts on $\mathbf{x}$ in $\mathbb{R}^n$ by

Every linear map is multiplication by exactly one matrix, whose $j$th column is the image of $\mathbf{e}_j$. The product $AB$ is defined when $A$ has as many columns as $B$ has rows, it applies $B$ first and then $A$, so $(AB)\mathbf{x} = A(B\mathbf{x})$, and in general $AB \ne BA$. A square matrix is invertible exactly when $\det A \ne 0$, and then $A\mathbf{x} = \mathbf{b}$ has the single solution $\mathbf{x} = A^{-1}\mathbf{b}$. For $ad - bc \ne 0$,

A $3 \times 3$ determinant expands along the first row as

The rules $\det(AB) = \det A \det B$ and $\det A^{T} = \det A$ hold for square matrices of one size. Swapping two rows changes the sign, scaling one row by $k$ scales the determinant by $k$, and adding a multiple of one row to another changes nothing. The same holds for columns.

• Area and volume. A $2 \times 2$ matrix multiplies every area by $\lvert \det A \rvert$, and a $3 \times 3$ matrix multiplies every volume by $\lvert \det A \rvert$.
• Orientation. An invertible map preserves orientation when $\det A > 0$ and reverses it when $\det A < 0$. An ordered basis $(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)$ is right-handed when $\mathbf{v}_1 \cdot (\mathbf{v}_2 \times \mathbf{v}_3) > 0$.
• Change of basis. If the columns of an invertible $P$ form a basis $\mathcal{B}$, the coordinates of $\mathbf{x}$ are $[\mathbf{x}]_{\mathcal{B}} = P^{-1}\mathbf{x}$, and the map with standard matrix $A$ has matrix $P^{-1}AP$ in those coordinates.

For example $A$ with rows $\langle 2, 1 \rangle$ and $\langle 1, 2 \rangle$ and $P$ with columns $\langle 1, 1 \rangle$ and $\langle 1, -1 \rangle$ give $P^{-1}AP = \operatorname{diag}(3, 1)$, so the map stretches by $3$ along $\langle 1, 1 \rangle$ and fixes $\langle 1, -1 \rangle$.

Lessons Matrices, linear maps and determinants

Lines and planes

The line through the point with position vector $\mathbf{r}_0 = \langle x_0, y_0, z_0 \rangle$ with direction $\mathbf{v} = \langle a, b, c \rangle \ne \mathbf{0}$ is

When $a$, $b$ and $c$ are all nonzero it has the symmetric form

A zero component drops its fraction, so $b = 0$ with $a, c \ne 0$ gives $y = y_0$ together with $\frac{x - x_0}{a} = \frac{z - z_0}{c}$. The plane through $\mathbf{r}_0$ with normal $\mathbf{n} = \langle a, b, c \rangle \ne \mathbf{0}$ is

with $d = \mathbf{n} \cdot \mathbf{r}_0$. Three points $P$, $Q$, $R$ not on one line lie in exactly one plane, with normal $\overrightarrow{PQ} \times \overrightarrow{PR}$.

Two lines are skew when they are not parallel and do not meet, which happens exactly when $\mathbf{v}_1 \times \mathbf{v}_2 \ne \mathbf{0}$ and $(\mathbf{r}_2 - \mathbf{r}_1) \cdot (\mathbf{v}_1 \times \mathbf{v}_2) \ne 0$. For example the point $(1, 2, 3)$ lies at distance $\frac{\lvert 2 - 2 + 6 - 1 \rvert}{3} = \frac{5}{3}$ from the plane $2x - y + 2z = 1$.

Lessons Equations of lines and planes

Cylinders and quadric surfaces

A trace is the intersection of a surface with a plane, found by fixing one coordinate. An equation that does not contain $z$ is a cylinder with rulings parallel to the $z$-axis, and the same holds for the other variables. With $a, b, c > 0$, the standard quadric surfaces with axis along the $z$-axis are

To identify an equation without cross terms, first look for a missing variable, which means a cylinder. Then complete the square in every variable that appears both squared and to the first power. With all three variables squared and the right side scaled to $1$, no minus sign on the left gives an ellipsoid, one gives a hyperboloid of one sheet and two give a hyperboloid of two sheets, and the axis belongs to the variable whose sign differs. With right side $0$ and mixed signs it is a cone. Two squared variables and one to the first power give a paraboloid, elliptic for equal signs and hyperbolic for opposite signs. For example $x^2 - 2x + y^2 - z^2 = 3$ becomes $(x - 1)^2 + y^2 - z^2 = 4$, a hyperboloid of one sheet whose axis is the line $x = 1$, $y = 0$.

Lessons Cylinders and quadric surfaces

Cylindrical and spherical coordinates

Cylindrical coordinates $(r, \theta, z)$ use polar coordinates in the $xy$-plane and keep the height,

with $r \ge 0$ the distance to the $z$-axis. Spherical coordinates $(\rho, \theta, \phi)$ use the distance $\rho$ to the origin, the same azimuth $\theta$, and the angle $\phi$ from the positive $z$-axis,

with $\rho \ge 0$, $0 \le \phi \le \pi$ and usually $0 \le \theta < 2\pi$. The two systems are linked by $r = \rho\sin\phi$ and $z = \rho\cos\phi$. Many physics books swap the letters $\theta$ and $\phi$, so check which angle a source means.

The formula $\theta = \arctan\frac{y}{x}$ gives the right direction only for $x > 0$, and a negative value then needs $2\pi$ added, so choose $\theta$ in the quadrant given by the signs of $x$ and $y$. The point $(-1, 1, \sqrt{6})$ has $r = \sqrt{2}$, $\theta = \frac{3\pi}{4}$, $\rho = 2\sqrt{2}$ and $\phi = \frac{\pi}{6}$, while $\arctan(-1) = -\frac{\pi}{4}$ would point the wrong way.

Off the $z$-axis the coordinates in these ranges are unique. On the $z$-axis the angle $\theta$ is arbitrary, and at the origin $\phi$ is arbitrary too.

Lessons Cylindrical and spherical coordinates

Vector-Valued Functions

A vector function traces a curve in space as its parameter runs, the way a moving object traces its path. Its calculus works one component at a time. From the derivatives of the position vector come tangent lines, the length of the curve, how sharply it bends and twists, and the velocity and acceleration of the motion.

Vector functions and their derivatives

A vector function $\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle$ has the intersection of the domains of its components as its natural domain. Limits, continuity, derivatives and integrals are all taken component by component, so

exists exactly when all three component derivatives exist. A continuous $\mathbf{r}$ on an interval traces a space curve, oriented in the direction of increasing $t$. The circular helix

with $a > 0$ and $b \ne 0$ lies on the cylinder $x^2 + y^2 = a^2$ and rises $2\pi b$ per turn. For $b > 0$ it turns counterclockwise as seen from above while it rises.

A parametrization on an interval $I$ is regular when it is $C^1$ on $I$, with one-sided derivatives at endpoints that belong to $I$, and $\mathbf{r}'(t) \ne \mathbf{0}$ for every $t$ in $I$. Where $\mathbf{r}'(t_0) \ne \mathbf{0}$, the tangent line and the unit tangent are

For example $\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle$ has $\mathbf{r}'(0) = \langle 0, 1, 1 \rangle$, so its tangent line at $(1, 0, 0)$ is $\langle 1, s, s \rangle$. For differentiable $\mathbf{u}$, $\mathbf{v}$ and $f$,

with the order of the factors kept in the cross product rule. The chain rule reads $\frac{d}{ds}\mathbf{u}(f(s)) = f'(s)\,\mathbf{u}'(f(s))$. If $\mathbf{r}$ is differentiable and $\lvert \mathbf{r}(t) \rvert$ is constant on an interval, then $\mathbf{r}'(t)$ is perpendicular to $\mathbf{r}(t)$. Integrals also go component by component, and for $\mathbf{r}$ continuous on $[a, b]$ with antiderivative $\mathbf{R}$,

Lessons Vector functions and space curves, Derivatives and integrals

Arc length and reparametrization

For $\mathbf{r}$ of class $C^1$ on $[a, b]$, the speed is $\lvert \mathbf{r}'(t) \rvert$ and the length is

For a piecewise $C^1$ parametrization, add the lengths of the pieces. The length counts multiplicity, so a curve traced twice has twice the length. The arc length function from $a$ is $s(t) = \int_a^t \lvert \mathbf{r}'(u) \rvert \, du$, with $\frac{ds}{dt} = \lvert \mathbf{r}'(t) \rvert$.

For a regular $\mathbf{r}$ the function $s$ is strictly increasing with a $C^1$ inverse, and $\mathbf{R}(s) = \mathbf{r}(t(s))$ has $\lvert \mathbf{R}'(s) \rvert = 1$, the parametrization by arc length. A reparametrization $\mathbf{r}(\varphi(u))$ with $\varphi$ of class $C^1$ and $\varphi' \ne 0$ keeps a regular curve regular and leaves its length unchanged. It preserves the orientation when $\varphi' > 0$ and reverses it when $\varphi' < 0$, which replaces $\mathbf{T}$ by $-\mathbf{T}$.

For example the helix $\langle 3\cos t, 3\sin t, 4t \rangle$ has speed $5$, length $10\pi$ over $0 \le t \le 2\pi$, and arc length parametrization

Lessons Arc length and reparametrization

Curvature, torsion and the moving frame

Each object needs its own hypotheses, and each row adds its condition to the rows above it.

The curvature measures turning per unit length,

For the graph $y = f(x)$ of a $C^2$ function this becomes $\kappa = \frac{\lvert f'' \rvert}{(1 + f'^2)^{3/2}}$. Where $\kappa \ne 0$, the principal unit normal and the binormal are

The frame $\mathbf{T}, \mathbf{N}, \mathbf{B}$ is right-handed and orthonormal. The osculating plane is spanned by $\mathbf{T}$ and $\mathbf{N}$, and the osculating circle lies in it with radius $\frac{1}{\kappa}$ and center $\mathbf{r} + \frac{1}{\kappa}\mathbf{N}$. The torsion is the signed number

and with this sign a helix that turns counterclockwise as it rises in right-handed coordinates has $\tau > 0$. The frame moves by the Frenet-Serret formulas

A curve with $\kappa \ne 0$ on an interval lies in one plane exactly when $\tau = 0$ there. The helix $\langle a\cos t, a\sin t, bt \rangle$ has

so $\langle 3\cos t, 3\sin t, 4t \rangle$ has $\kappa = \frac{3}{25}$ and $\tau = \frac{4}{25}$.

Lessons Curvature, torsion and the moving frame

Velocity and acceleration

For a position $\mathbf{r}(t)$ the velocity is $\mathbf{v} = \mathbf{r}'$, the speed is $v = \lvert \mathbf{v} \rvert$ and the acceleration is $\mathbf{a} = \mathbf{r}''$. For $\mathbf{r}$ of class $C^2$ with $\mathbf{v} \ne \mathbf{0}$,

where the $\mathbf{N}$ term appears only when $\kappa \ne 0$. Also $a_T^2 + a_N^2 = \lvert \mathbf{a} \rvert^2$, and the acceleration has no binormal part.

A projectile under gravity alone, with no air resistance, launched from height $h \ge 0$ at speed $v_0$ and angle $\alpha$ above the horizontal, has

until it lands. For $h = 0$ and $0 < \alpha < \frac{\pi}{2}$, the flight time, range and maximum height are

With a launch speed of $20$ meters per second, $\alpha = \frac{\pi}{6}$ and $g = 9.8$ meters per second squared, these are about $2.04$ seconds, $35.3$ meters and $5.10$ meters.

For a particle of constant mass $m$ the angular momentum $\mathbf{L} = \mathbf{r} \times m\mathbf{v}$ has derivative $\mathbf{r} \times \mathbf{F}$. A central force, parallel to $\mathbf{r}$, keeps $\mathbf{L}$ constant, and if $\mathbf{L} \ne \mathbf{0}$ the motion stays in one plane and sweeps area at the rate $\frac{\lvert \mathbf{L} \rvert}{2m}$. For a planet moving only under the inverse-square gravity of a sun of mass $M$ fixed at the origin, with $\mathbf{L} \ne \mathbf{0}$, Kepler’s laws hold. The orbit is a conic with a focus at the sun, an ellipse when bounded, equal areas are swept in equal times, and an elliptical orbit with semimajor axis $a$ has period $T$ with

Lessons Velocity and acceleration in space

Differentiation of Functions of Several Variables

A function of several variables can change in infinitely many directions at once. This unit defines limits and derivatives that account for every direction, organizes the derivative as a gradient or a Jacobian matrix, and uses it to approximate functions, to solve equations locally and to find largest and smallest values.

Functions, domains and level sets

When only a formula is given, the domain is the largest set where it gives a real number. The graph of $f(x, y)$ is the surface $z = f(x, y)$, the level curve at level $c$ is the set where $f(x, y) = c$, and for three variables the level surface is the set where $F(x, y, z) = k$. A contour map draws level curves for equally spaced values. For example $f(x, y) = \sqrt{9 - x^2 - y^2}$ has the closed disk $x^2 + y^2 \le 9$ as its domain, the range $[0, 3]$, and circles centered at the origin as level curves for $0 \le c < 3$, while the level curve for $c = 3$ is the origin alone.

A point $P$ is an interior point of a set $S$ when some open ball about $P$ lies in $S$, and a boundary point when every open ball about $P$ meets both $S$ and its complement. The set is open when all its points are interior, closed when it contains all its boundary points, bounded when it lies in some ball, and compact when it is closed and bounded. Many sets are neither open nor closed.

Lessons Functions of several variables

Limits and continuity

Let $(a, b)$ be a limit point of the domain $D$ of $f$. Then $f(x, y) \to L$ as $(x, y) \to (a, b)$ when for every $\varepsilon > 0$ there is $\delta > 0$ with $\lvert f(x, y) - L \rvert < \varepsilon$ for every $(x, y)$ in $D$ with

• Paths can disprove a limit. Two paths in $D$ toward $(a, b)$ with different limits show that the limit does not exist. Along $y = mx$ the function $\frac{xy}{x^2 + y^2}$ tends to $\frac{m}{1 + m^2}$, which depends on $m$.
• Paths cannot prove a limit. The function $\frac{x^2 y}{x^4 + y^2}$ tends to $0$ along every line through the origin, but equals $\frac{1}{2}$ along $y = x^2$, so it has no limit at the origin.
• Squeeze by distance. If $\lvert f(x, y) - L \rvert \le g(x, y)$ near $(a, b)$ and $g \to 0$, then $f \to L$. Since $x^2 \le x^2 + y^2$, the bound $\left\lvert \frac{x^2 y}{x^2 + y^2} \right\rvert \le \lvert y \rvert \le r$ shows that this function tends to $0$.

The function $f$ is continuous at a point $(a, b)$ of its domain when the limit there equals $f(a, b)$, with only points of $D$ tested. Sums, products, quotients with nonzero denominator and compositions of continuous functions are continuous, so polynomials are continuous everywhere and rational functions on their domains.

Lessons Multivariable limits and continuity

Partial derivatives

At an interior point $(a, b)$ of the domain, the partial derivatives vary one coordinate and hold the other fixed,

They are the slopes of the traces in the planes $y = b$ and $x = a$. In the subscript notation the order of differentiation reads from left to right, and in the $\partial$ notation from right to left,

Clairaut’s theorem. If $f_{xy}$ and $f_{yx}$ exist and are continuous on an open disk containing $(a, b)$, then $f_{xy}(a, b) = f_{yx}(a, b)$. More generally, for $f$ of class $C^k$ on an open set, every partial derivative of order at most $k$ is independent of the order of differentiation. For example $f = x^3 y^2 + e^{xy}$ has $f_x = 3x^2 y^2 + y e^{xy}$ and $f_{xy} = f_{yx} = 6x^2 y + (1 + xy)e^{xy}$. The existence of both partial derivatives at a point does not even make $f$ continuous there.

The classical partial differential equations, with constants $\alpha \ne 0$ and $c \ne 0$, are Laplace’s equation $u_{xx} + u_{yy} = 0$, the heat equation $u_t = \alpha^2 u_{xx}$ and the wave equation $u_{tt} = c^2 u_{xx}$.

Lessons Partial derivatives

Differentiability and tangent planes

Let $U$ be open in $\mathbb{R}^n$, $f$ map $U$ to $\mathbb{R}^m$ and $\mathbf{a}$ lie in $U$. The function is differentiable at $\mathbf{a}$ when there is a linear map $L$, the total derivative $Df(\mathbf{a})$, with

Then every partial derivative exists at $\mathbf{a}$, $Df(\mathbf{a})$ is multiplication by the $m \times n$ Jacobian matrix $\left[ \frac{\partial f_i}{\partial x_j}(\mathbf{a}) \right]$, with row $i$ for the component $f_i$, and $f$ is continuous at $\mathbf{a}$. The converse fails. Partial derivatives can exist where $f$ is not differentiable, but if all partial derivatives exist near $\mathbf{a}$ and are continuous at $\mathbf{a}$, then $f$ is differentiable at $\mathbf{a}$. In particular every $C^1$ function on an open set is differentiable there.

For $f(x, y)$ differentiable at $(a, b)$, the linearization, whose graph is the tangent plane, is

with normal vector $\langle f_x(a, b), f_y(a, b), -1 \rangle$. The total differential $dz = f_x\,dx + f_y\,dy$ approximates the actual change $\Delta z$ with an error that is small compared to $\sqrt{dx^2 + dy^2}$. For example $f = x^2 + 3xy$ at $(1, 2)$ has $f = 7$, $f_x = 8$ and $f_y = 3$, so $f(1.1, 1.9) \approx 7 + 0.8 - 0.3 = 7.5$, against the exact value $7.48$. The function equal to $\frac{xy}{x^2 + y^2}$ away from the origin and $0$ at the origin has $f_x(0, 0) = f_y(0, 0) = 0$ but is not continuous there, so it has no tangent plane at the origin.

Lessons Total derivative, tangent planes and differentials

The chain rule

If $g$ is differentiable at $\mathbf{a}$ and $f$ is differentiable at $g(\mathbf{a})$, then $f \circ g$ is differentiable at $\mathbf{a}$ and