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.
01
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.
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.
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.
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.
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.
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.
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.
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.
