What does this Multivariable Calculus cheat sheet cover?

It has 10 sections: Vectors, Lines & Planes, Partial Derivs, Chain Rule, Optimization, Double/Triple Int, Coordinates, Line Integrals, Green & Stokes and Divergence. Together they cover the topics a typical Multivariable Calculus exam tests, including vectors, dot and cross product, lines and planes, partial derivatives and gradient and directional derivatives.

Is the multivariable calculus cheat sheet free?

The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.

Can I bring this Multivariable Calculus cheat sheet to my exam?

Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.

The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 2 statements before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.

Can I make a Multivariable Calculus sheet from my own notes?

Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

MathMultivariable Calculus

Multivariable Calculus Cheat Sheet — 1-page PDF

Vectors, derivatives, integrals, coordinates, and vector calculus theorems. The whole course on one dense, printable page for your midterm or final, with every formula typeset and worked examples where they help.

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For a specific exam: Calculus III (Multivariable) Final exam study sheet.

1 pageUS Letter
10 sections11 topics
Free previewfull sheet with Unlimited

Preview of the one-page Multivariable Calculus cheat sheet: Vectors, derivatives, integrals, coordinates, and vector calculus theorems — the whole course, one page

On the sheet

What’s on it

The topics a typical Multivariable Calculus exam tests, all on the one page.

Vectors, dot and cross product
Lines and planes
Partial derivatives and gradient
Directional derivatives
Tangent planes
Chain rule multivariable
Lagrange multipliers
Double and triple integrals
Polar, cylindrical, spherical coordinates and Jacobians
Line integrals
Green's, Stokes', divergence theorems

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The free part, as text

The free part of the sheet, section by section. Unlock the full sheet to read the rest, edit any block, and print it or save it as a PDF.

01Vectors

Vector: Quantity with magnitude + direction; in $R^{n}$ , $v = ⟨ v_{1}, \dots, v_{n} ⟩$ .

Magnitude

∥ v ∥ = v_{1}^{2} + \dots + v_{n}^{2}

Euclidean norm

Unit vector

u = v /∥ v ∥

if $v \neq = 0$

Dot product

u \cdot v = u_{1} v_{1} + \dots + u_{n} v_{n} = ∥ u ∥ ∥ v ∥ cos θ

$θ$ = angle between vectors

Orthogonal

u \cdot v = 0

iff vectors are perpendicular

Projection

proj_{u} v = \frac{v \cdot u}{u \cdot u} u

vector projection of $v$ onto $u$

Cross product

u \times v = i u_{1} v_{1} j u_{2} v_{2} k u_{3} v_{3}

in $R^{3}$ only

Cross magnitude

∥ u \times v ∥ = ∥ u ∥ ∥ v ∥ sin θ

area of parallelogram

Dot vs cross

Operation	Output	Use
$\cdot$	scalar	angle / proj / orthogonality
$\times$	vector	normal vector / area / torque

02Partial Derivs

Directional derivative

D_{u} f = \nabla f \cdot u

$u$ must be unit vector

Max rate

max D_{u} f = ∥\nabla f ∥

at direction $u = \nabla f /∥\nabla f ∥$

Tangent to level set

\nabla f ⊥ level curve/surface f = c

at points where $\nabla f \neq = 0$

Ex.If $f (x, y) = x^{2} y + sin y$ , find $f_{x}, f_{y}$ and $D_{u} f$ at $(1, 0)$ for $u = ⟨ 3/5, 4/5 ⟩$ .

$f_{x} = 2 x y$ , $f_{y} = x^{2} + cos y$ . At $(1, 0)$ , $\nabla f = ⟨ 0, 2 ⟩$ . So $D_{u} f = ⟨ 0, 2 ⟩ \cdot ⟨ 3/5, 4/5 ⟩ = 8/5$ .

03Chain Rule

2-var chain

\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t}

if $z = f (x, y)$ and $x, y$ depend on $t$

Multivar chain

\frac{\partial z}{\partial s} = z_{x} x_{s} + z_{y} y_{s}, \frac{\partial z}{\partial t} = z_{x} x_{t} + z_{y} y_{t}

for $z = f (x, y)$ with $x (s, t), y (s, t)$

Gradient form

\frac{d}{d t} f (r (t)) = \nabla f (r (t)) \cdot r^{'} (t)

composition with a curve

Tangent plane

z - f (a, b) = f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)

for $z = f (x, y)$

04Double/Triple Int

Double integral

\iint_{R} f (x, y) d A

volume / mass / average over region $R$

Fubini

\iint_{R} f (x, y) d A = \int_{a}^{b} \int_{g_{1} (x)}^{g_{2} (x)} f (x, y) d y d x

for Type I region; reverse order similarly

Volume

V = \iint_{R} 1 d A = ∭_{E} 1 d V

over planar region $R$ or solid $E$

Triple integral

∭_{E} f (x, y, z) d V

mass if $f = ρ$

Center of mass

\overset{x}{ˉ} = \frac{1}{M} ∭_{E} x ρ d V, \overset{y}{ˉ} = \frac{1}{M} ∭_{E} y ρ d V, \overset{z}{ˉ} = \frac{1}{M} ∭_{E} z ρ d V

$M = ∭_{E} ρ d V$

Average value

f_{avg} = \frac{1}{Area ( R )} \iint_{R} f d A

same for volume with $∭$

Watch out:Order matters in iterated integrals only through limits; integrand unchanged. Sketch region before switching order.

05Coordinates

Coordinate systems

System	Change of vars	Jacobian
Polar	$x = r cos θ, y = r sin θ$	$d A = r d r d θ$
Cylindrical	$x = r cos θ, y = r sin θ, z = z$	$d V = r d r d θ d z$
Spherical	$x = ρ sin ϕ cos θ, y = ρ sin ϕ sin θ, z = ρ cos ϕ$	$d V = ρ^{2} sin ϕ d ρ d ϕ d θ$

06Line Integrals

Path independence tests

Condition	If simply connected
$F = \nabla f$	path independent
$\nabla \times F = 0$	conservative

Watch out:For vector line integrals, use $d r = r^{'} (t) d t$ , not $d s$ . For scalar line integrals, use $d s$ .

07Green & Stokes

Green's theorem

\oint_{C} P d x + Q d y = \iint_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d A

$C$ positively oriented, simple closed curve

Circulation form

\oint_{C} F \cdot d r = \iint_{D} (\nabla \times F) \cdot k d A

with $F = ⟨ P, Q ⟩$

Flux form

\oint_{C} F \cdot n d s = \iint_{D} \nabla \cdot F d A

2D flux across $C$

Stokes' theorem

\oint_{\partial S} F \cdot d r = \iint_{S} (\nabla \times F) \cdot n d S

boundary orientation by right-hand rule

Curl

\nabla \times F = i \partial_{x} P j \partial_{y} Q k \partial_{z} R

for $F = ⟨ P, Q, R ⟩$

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Also on the full Multivariable Calculus sheet

8 more sections, with every definition, formula and worked example. Unlimited unlocks every full sheet in the library to edit, print or save as a PDF, and makes unlimited sheets from your own notes.

Lines & Planes

Line
Parametric line
Plane
Scalar plane
Plane from 3 pts
Distance point-plane
Distance point-line

Partial Derivs

Partial derivative
Notation
Clairaut
Gradient

Chain Rule

Linearization
Differential

Optimization

Critical point
Second derivative test
Classification
Lagrange multiplier
Constrained system
Two constraints

Coordinates

Polar Jacobian
Cylindrical Jacobian
Spherical Jacobian
Spherical relations

Line Integrals

Scalar line integral
Vector line integral
Arc length
Conservative field
Fundamental theorem

Green & Stokes

Divergence

Divergence
Divergence theorem
Flux
Vector calculus theorem map

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About the Multivariable Calculus sheet and making your own.

What does this Multivariable Calculus cheat sheet cover?: It has 10 sections: Vectors, Lines & Planes, Partial Derivs, Chain Rule, Optimization, Double/Triple Int, Coordinates, Line Integrals, Green & Stokes and Divergence. Together they cover the topics a typical Multivariable Calculus exam tests, including vectors, dot and cross product, lines and planes, partial derivatives and gradient and directional derivatives.
Is the multivariable calculus cheat sheet free?: The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.
Can I bring this Multivariable Calculus cheat sheet to my exam?: Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.
How accurate is it?: The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 2 statements before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.
Can I make a Multivariable Calculus sheet from my own notes?: Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

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