Multivariable calculus: geometry and theorem choice
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 180 minutes, as an unvalidated planning estimate. This collection contains 10 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Choose and justify a method before opening the solution. Include the assumptions, intermediate work and interpretation that make the answer defensible.
These public mixed tasks sample the unit; they are not a secure exam, independent certification or a guarantee of AP or university readiness.
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0 of 10 tasks have recorded responses. No overall mastery score is assigned.
For r(t)=(2t,t²,1−t), find velocity, acceleration and speed at t=1. Find the tangent line there and the plane through r(1) perpendicular to that line. Explain why the plane normal is not the acceleration.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Define g(x,y)=xy²/(x²+y⁴) away from the origin and g(0,0)=0. Test continuity and coordinate partial derivatives at the origin. Separately linearize f(x,y)=x²+3xy+y² at (1,2), predict its change for displacement (0.02,−0.01), and give the exact remainder.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Find and classify all stationary points of f=x²+2y²−4x+8y. Then find the minimum and maximum of h=xy on x²+y²=8 using multipliers, and justify that no extrema were omitted.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A lamina occupies 0≤x≤2, x≤y≤2, with density ρ(x,y)=y. Set up mass in both orders and calculate mass and its center of mass. State units if x,y are lengths and ρ is mass per area.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Find ∫∫D(x²+y²)dA for the annulus 1≤x²+y²≤9. Set up and evaluate the volume of the ball x²+y²+z²≤16 in spherical coordinates. Separately transform the rectangle 0≤u≤1,0≤v≤2 by x=2u+v,y=u−v and find its image area. Explain the three volume/area factors.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Along r(t)=(t,2t), 0≤t≤1, compute ∫C(x+y)ds and the work of F=(2xy,x²). Find a potential for F and check the work from endpoints. Explain the effect of reversing the path on each integral.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For F=(−2y/(x²+y²),2x/(x²+y²)), calculate circulation around the circle of radius 3 counterclockwise. Its scalar curl is zero wherever defined. Explain why Green’s theorem does not give zero on the whole disk, and what it says for the annulus 9≤x²+y²≤25 with positive boundary orientation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Parameterize the graph z=2x+y over 0≤x≤1,0≤y≤2. Find its area and upward flux of F=(0,0,z). Explain why replacing the oriented area vector by its length would give the wrong flux integrand.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For F=(−y,z,x), find circulation around x²+y²=4,z=0 counterclockwise as seen from above, using Stokes and also a direct parameterization. State regularity and orientation conditions.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For F=(x,y,2z), find outward flux through the closed cylinder x²+y²≤9,0≤z≤2. Split the result into top, bottom and side flux to verify it. Explain why applying the volume formula to the side alone is invalid.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
For r(t)=(2t,t²,1−t), find velocity, acceleration and speed at t=1. Find the tangent line there and the plane through r(1) perpendicular to that line. Explain why the plane normal is not the acceleration.
Define g(x,y)=xy²/(x²+y⁴) away from the origin and g(0,0)=0. Test continuity and coordinate partial derivatives at the origin. Separately linearize f(x,y)=x²+3xy+y² at (1,2), predict its change for displacement (0.02,−0.01), and give the exact remainder.
Find and classify all stationary points of f=x²+2y²−4x+8y. Then find the minimum and maximum of h=xy on x²+y²=8 using multipliers, and justify that no extrema were omitted.
A lamina occupies 0≤x≤2, x≤y≤2, with density ρ(x,y)=y. Set up mass in both orders and calculate mass and its center of mass. State units if x,y are lengths and ρ is mass per area.
Find ∫∫D(x²+y²)dA for the annulus 1≤x²+y²≤9. Set up and evaluate the volume of the ball x²+y²+z²≤16 in spherical coordinates. Separately transform the rectangle 0≤u≤1,0≤v≤2 by x=2u+v,y=u−v and find its image area. Explain the three volume/area factors.
Along r(t)=(t,2t), 0≤t≤1, compute ∫C(x+y)ds and the work of F=(2xy,x²). Find a potential for F and check the work from endpoints. Explain the effect of reversing the path on each integral.
For F=(−2y/(x²+y²),2x/(x²+y²)), calculate circulation around the circle of radius 3 counterclockwise. Its scalar curl is zero wherever defined. Explain why Green’s theorem does not give zero on the whole disk, and what it says for the annulus 9≤x²+y²≤25 with positive boundary orientation.
Parameterize the graph z=2x+y over 0≤x≤1,0≤y≤2. Find its area and upward flux of F=(0,0,z). Explain why replacing the oriented area vector by its length would give the wrong flux integrand.
For F=(−y,z,x), find circulation around x²+y²=4,z=0 counterclockwise as seen from above, using Stokes and also a direct parameterization. State regularity and orientation conditions.
For F=(x,y,2z), find outward flux through the closed cylinder x²+y²≤9,0≤z≤2. Split the result into top, bottom and side flux to verify it. Explain why applying the volume formula to the side alone is invalid.
