Learn Multivariable Calculus unit by unit

Analyze functions, derivatives, and integrals in higher dimensions.

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Units at a glance

Every unit in this course, what you will be able to do once you finish it, and roughly when it lands in a 14-week schedule.

Multivariable Calculus course map: 7 units
#	Unit	What you can do after it	Weeks
01	Vectors and space	You can add vectors, take dot and cross products, and describe lines and planes in three dimensions.	Weeks 1-2
02	Partial derivatives	You can differentiate a function of several variables one variable at a time and collect them into a gradient.	Weeks 3-4
03	Multiple integrals	You can integrate over a flat region or a solid and choose the order that makes the limits simple.	Weeks 5-6
04	Vector fields	You can picture a field of arrows, compute its divergence and curl, and test whether it comes from a potential function.	Weeks 7-8
05	Line integrals	You can add a quantity up along a curved path and compute the work a force does along it.	Weeks 9-10
06	Surface integrals	You can integrate over a curved surface and compute the flux passing through it.	Weeks 11-12
07	Green, Stokes, and divergence theorems	You can trade a hard boundary integral for an easier region integral, or run the swap the other way.	Weeks 13-14

Multivariable Calculus course units

Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.

Vectors and space

Weeks 1-2

You can add vectors, take dot and cross products, and describe lines and planes in three dimensions. You can also find an angle or a distance in space.

Find the angle between two vectors from their dot product.
Use a cross product to get a vector at right angles to two others.
Write a plane's equation from one point and a normal vector.

Worked example

Find the dot product of (2, 1, -3) and (4, 0, 5)

2(4) + 1(0) + (-3)(5)
8 + 0 - 15

Answer-7

Most common mistakeMultiplying component by component and keeping a vector: writing (8, 0, -15) instead of the single number -7, so the angle formula then fails.

Explain vectors and space Practice vectors and space

Partial derivatives

Weeks 3-4

You can differentiate a function of several variables one variable at a time and collect them into a gradient. You can also find the direction of steepest climb on a surface.

Differentiate with respect to x while holding y fixed as a constant.
Compute a directional derivative from the gradient and a unit vector.
Write the tangent plane to a surface at a point.

Worked example

For f(x, y) = x squared y + 3y squared, find both partial derivatives

With y held fixed, the 3y squared term is constant: df/dx = 2xy
With x held fixed: df/dy = x squared + 6y

Answerdf/dx = 2xy and df/dy = x squared + 6y

Most common mistakeDifferentiating both variables at once: writing df/dx = 2xy + 6y, which wrongly differentiates the 3y squared term while x is the active variable.

Explain partial derivatives Practice partial derivatives

Multiple integrals

Weeks 5-6

You can integrate over a flat region or a solid and choose the order that makes the limits simple. You can also switch to polar or cylindrical coordinates.

Set the limits for a double integral over a triangle.
Swap the order of integration when the inner integral is impossible.
Convert a circular region to polar and keep the extra factor of r.

Worked example

Evaluate the double integral of xy, with y from 0 to 2 and x from 0 to 1

Inner integral in y: x times y squared / 2, from 0 to 2, gives 2x
Outer integral in x: the integral of 2x from 0 to 1

Answer1

Most common mistakeDropping the extra r when converting to polar: the unit disk's area then comes out as 2 pi rather than pi.

Explain multiple integrals Practice multiple integrals

Vector fields

Weeks 7-8

You can picture a field of arrows, compute its divergence and curl, and test whether it comes from a potential function. You can also sketch which way the flow runs.

Compute the divergence of a two-part field.
Test a flat field for being conservative by comparing two mixed partials.
Find a potential function once you know one exists.

Worked example

Is F = (2xy, x squared) conservative?

P = 2xy, so dP/dy = 2x
Q = x squared, so dQ/dx = 2x

AnswerThey match, so yes, and a potential is f = x squared y

Most common mistakeComparing dP/dx with dQ/dy instead: 2y against 0 never match, so a genuinely conservative field gets rejected.

Explain vector fields Practice vector fields

Line integrals

Weeks 9-10

You can add a quantity up along a curved path and compute the work a force does along it. You can also skip the path entirely when the field is conservative.

Parametrize a path and substitute it into the integral.
Compute work as the integral of F dotted with dr.
Use the fundamental theorem for line integrals when a potential exists.

Worked example

F = (2xy, x squared) has potential f = x squared y. Find the work from (0,0) to (2,3).

Work equals the potential at the end minus at the start
f(2, 3) = 4 x 3 = 12 and f(0, 0) = 0

Answer12

Most common mistakeReporting different answers for different routes between the same two points: for a conservative field every route gives 12, so a mismatch means a slip in the parametrization.

Explain line integrals Practice line integrals

Surface integrals

Weeks 11-12

You can integrate over a curved surface and compute the flux passing through it. You can also fix an outward direction at the start and stay with it.

Set up the surface element for a surface written as z = g(x, y).
Compute flux as the integral of F dotted with the unit normal.
Choose and state the outward normal on a closed surface.

Worked example

Find the flux of F = (0, 0, 3) upward through the disk of radius 3 in the plane z = 1

The upward unit normal is (0, 0, 1), so F dotted with it is 3
The field is constant, so flux is 3 times the disk's area

Answer3 x 9 pi = 27 pi

Most common mistakeUsing the circle's circumference in place of its area: computing 3 x 2 pi x 3 = 18 pi instead of 27 pi.

Explain surface integrals Practice surface integrals

Green, Stokes, and divergence theorems

Weeks 13-14

You can trade a hard boundary integral for an easier region integral, or run the swap the other way. You can also check that the region meets each theorem's conditions first.

Turn a closed line integral into a double integral with Green's theorem.
Apply the divergence theorem to a closed surface wrapped around a solid.
Confirm the boundary is closed and positively oriented before you use a theorem.

Worked example

Use Green's theorem on the loop integral of (-y dx + x dy) around the unit circle

dQ/dx - dP/dy = 1 - (-1) = 2
Integrate the constant 2 over the unit disk: 2 times pi

Answer2 pi

Most common mistakeApplying Green's theorem to a path that does not close: the region integral then counts area the curve never actually enclosed.

Explain green, stokes, and divergence theorems Practice green, stokes, and divergence theorems

Prepare for Multivariable Calculus practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Vectors and space from a blank page. Then connect it to Partial derivatives. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.

Use tools to verify, not to choose the method for you

The 3D calculator can test calculations and representations used in Multivariable Calculus. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.

Know when to move to the full test

Move from Multivariable Calculus practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.

Before and after the syllabus

Learn the ideas, then practise them

The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.

GuidesCalculus explainedConcepts, worked methods and checks behind this course.

PracticeMixed Calculus questionsAnswer review after every attempt.

ReferenceFormulas for this courseConditions and a worked example for each one.

Assess

Take the complete Multivariable Calculus test

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Reference

Review essential formulas

Open library →

Calculate

Use the 3D calculator

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Plan

Prepare around your exam date

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Questions about the Multivariable Calculus course

Where should I start?

Start with Vectors and space if you are following the full sequence. If that unit feels automatic, open the Multivariable Calculus practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.

How do I know I am ready for the course test?

You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Multivariable Calculus test only after you can correct practice errors from a blank page.

Which calculator supports this course?

The 3D calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.

Learn Multivariable Calculus unit by unit

Units at a glance

Multivariable Calculus course units

Vectors and space

Worked example

Partial derivatives

Worked example

Multiple integrals

Worked example

Vector fields

Worked example

Line integrals

Worked example

Surface integrals

Worked example

Green, Stokes, and divergence theorems

Worked example

Prepare for Multivariable Calculus practice

Start with the earliest uncertain prerequisite

Use tools to verify, not to choose the method for you

Know when to move to the full test

Learn the ideas, then practise them

Other university courses

Linear Algebra

Differential Equations

Discrete Mathematics

Take the complete Multivariable Calculus test

Review essential formulas

Use the 3D calculator

Prepare around your exam date

Questions about the Multivariable Calculus course