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← Multivariable Calculus
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Practice deliberately

Work through Multivariable Calculus unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 7 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

Multivariable Calculus coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the Multivariable Calculus syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use calculus practice questions with answers.

01

Vectors and space

Track dimensions and geometric meaning while computing, then verify the result by multiplication, substitution, or a basis-vector test.

Goal: recognise a vectors and space problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Partial derivatives

Relate local change to slope, apply the appropriate derivative rule, and use sign, units, and graph behavior to interpret the result.

Goal: recognise a partial derivatives problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Multiple integrals

Represent accumulated change, choose bounds and technique carefully, and check the result with units, area, or differentiation.

Goal: recognise a multiple integrals problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Vector fields

Track dimensions and geometric meaning while computing, then verify the result by multiplication, substitution, or a basis-vector test.

Goal: recognise a vector fields problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Line integrals

Represent accumulated change, choose bounds and technique carefully, and check the result with units, area, or differentiation.

Goal: recognise a line integrals problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Surface integrals

Represent accumulated change, choose bounds and technique carefully, and check the result with units, area, or differentiation.

Goal: recognise a surface integrals problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Green, Stokes, and divergence theorems

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a green, stokes, and divergence theorems problem from its wording, carry out the governing method, and check that the result is reasonable.

Multivariable Calculus questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 7 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: For u = (2, -1, 3) and v = (1, 4, -2), compute the dot product and the cross product u x v.

Multivariable Calculus · Vectors and space

  1. dot -8, cross (-10, 7, 9)
  2. dot 8, cross (-10, 7, 9)
  3. dot -8, cross (10, -7, -9)
  4. dot -8, cross (-10, -7, 9)
Show the worked answer
  1. Dot product: 2(1) + (-1)(4) + 3(-2) = 2 - 4 - 6 = -8.
  2. First cross component: (-1)(-2) - (3)(4) = 2 - 12 = -10.
  3. Second component (remember the minus sign): -[(2)(-2) - (3)(1)] = -(-4 - 3) = 7.
  4. Third component: (2)(4) - (-1)(1) = 8 + 1 = 9. So u x v = (-10, 7, 9).

Answer: dot -8, cross (-10, 7, 9)

Question 2: What is the distance from the point (3, 1, 2) to the plane 2x - y + 2z = 6?

Multivariable Calculus · Vectors and space

  1. 1
  2. 3
  3. 1/3
  4. 9
Show the worked answer
  1. Distance is |ax + by + cz - d| divided by the length of the normal vector (a, b, c).
  2. Top: |2(3) - 1(1) + 2(2) - 6| = |6 - 1 + 4 - 6| = 3.
  3. The normal is (2, -1, 2), with length the square root of 4 + 1 + 4 = 3.
  4. 3 / 3 = 1.

Answer: 1

Question 3: For f(x, y) = x^3 y^2 - 4xy, what is the mixed partial derivative f_xy at (2, 1)?

Multivariable Calculus · Partial derivatives

  1. 20
  2. 8
  3. 24
  4. -4
Show the worked answer
  1. First take the partial with respect to x: f_x = 3x^2 y^2 - 4y.
  2. Now take the partial of that with respect to y: f_xy = 6x^2 y - 4.
  3. At (2, 1): 6(4)(1) = 24.
  4. 24 - 4 = 20.

Answer: 20

Question 4: Evaluate the double integral of (x + 2y) over the rectangle 0 to 2 in x and 0 to 3 in y.

Multivariable Calculus · Multiple integrals

  1. 24
  2. 18
  3. 30
  4. 12
Show the worked answer
  1. Integrate with respect to y first: the integral of x + 2y dy is xy + y^2.
  2. Evaluate from y = 0 to 3: 3x + 9.
  3. Now integrate 3x + 9 from x = 0 to 2: 1.5x^2 + 9x.
  4. At x = 2 that is 6 + 18 = 24.

Answer: 24

Question 5: Is F = (2xy, x^2 + 3y^2) conservative, and if so what is a potential function?

Multivariable Calculus · Vector fields

  1. Yes, f = x^2 y + y^3
  2. No, the field is not conservative
  3. Yes, f = 2x^2 y + 3y^3
  4. Yes, f = x^2 y + 3y^2
Show the worked answer
  1. Check the cross partials. The partial of 2xy with respect to y is 2x.
  2. The partial of x^2 + 3y^2 with respect to x is also 2x, so the field is conservative.
  3. Integrate the first part with respect to x: x^2 y plus a function of y.
  4. Differentiating x^2 y + g(y) with respect to y and matching x^2 + 3y^2 gives g'(y) = 3y^2, so g = y^3 and f = x^2 y + y^3.

Answer: Yes, f = x^2 y + y^3

Question 6: Using F = (2xy, x^2 + 3y^2), what is the work done along any path from (0, 0) to (1, 2)?

Multivariable Calculus · Line integrals

  1. 10
  2. 2
  3. 8
  4. it depends on the path
Show the worked answer
  1. The field is conservative with potential f = x^2 y + y^3, so only the endpoints matter.
  2. At (1, 2): f = 1(2) + 2^3 = 2 + 8 = 10.
  3. At (0, 0): f = 0.
  4. The work is 10 - 0 = 10.

Answer: 10

Question 7: Use the divergence theorem to find the flux of F = (x, y, z) out of the sphere of radius 2.

Multivariable Calculus · Green, Stokes, and divergence theorems

  1. 32 pi
  2. 16 pi
  3. 8 pi
  4. 96 pi
Show the worked answer
  1. The divergence of (x, y, z) is 1 + 1 + 1 = 3.
  2. The flux equals the divergence integrated over the enclosed volume.
  3. The sphere of radius 2 has volume (4/3) pi (2^3) = 32 pi / 3.
  4. 3 x 32 pi / 3 = 32 pi.

Answer: 32 pi

Question 8: Use Green's theorem to evaluate the line integral of y dx + 2x dy counterclockwise around a circle of radius 3.

Multivariable Calculus · Surface integrals

  1. 9 pi
  2. 3 pi
  3. 18 pi
  4. 6 pi
Show the worked answer
  1. Green's theorem turns the loop integral into a double integral of (partial of 2x in x) minus (partial of y in y).
  2. That is 2 - 1 = 1.
  3. So the answer is just the area of the region.
  4. The circle of radius 3 has area pi(3^2) = 9 pi.

Answer: 9 pi

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed Multivariable Calculus questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Vectors and spaceFor u = (2, -1, 3) and v = (1, 4, -2), compute the dot product and the cross product u x v.dot -8, cross (-10, 7, 9)
2Vectors and spaceWhat is the distance from the point (3, 1, 2) to the plane 2x - y + 2z = 6?1
3Partial derivativesFor f(x, y) = x^3 y^2 - 4xy, what is the mixed partial derivative f_xy at (2, 1)?20
4Multiple integralsEvaluate the double integral of (x + 2y) over the rectangle 0 to 2 in x and 0 to 3 in y.24
5Vector fieldsIs F = (2xy, x^2 + 3y^2) conservative, and if so what is a potential function?Yes, f = x^2 y + y^3
6Line integralsUsing F = (2xy, x^2 + 3y^2), what is the work done along any path from (0, 0) to (1, 2)?10
7Green, Stokes, and divergence theoremsUse the divergence theorem to find the flux of F = (x, y, z) out of the sphere of radius 2.32 pi
8Surface integralsUse Green's theorem to evaluate the line integral of y dx + 2x dy counterclockwise around a circle of radius 3.9 pi

How the generated Multivariable Calculus sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this Multivariable Calculus practice page

What does this page cover?

It covers all 7 Multivariable Calculus units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.