Calculus III (Multivariable)

Use this for “Where does the line through (1, 2, −1) along ⟨2, 0, 3⟩ meet the plane x + y + z = 10?”

Use this for “For r(t) = ⟨3 cos t, 3 sin t, 4t⟩, find the velocity at t = 0, the speed, the length of one turn and the curvature.”

Use this for “Name the surface x² + y² − z² = 1.”

Use this for “How fast does f(x, y) = x² + xy + 2y² change at (1, 2) in the direction of ⟨3, 4⟩?”

Use this for “At (1, 2), ∂z/∂x = 4 and ∂z/∂y = 9. If x = 1 + 2t and y = 2 − t, find dz/dt at t = 0.”

Use this for “Find the critical point of f(x, y) = x² + xy + y² − 3x and say whether it is a maximum, a minimum or a saddle point.”

Use this for “Find the largest value of xy on the line 2x + 4y = 400, using a Lagrange multiplier.”

Use this for “Find ∬ x dA over the region between y = x² and y = 3x.”

Use this for “Evaluate ∬ (x² + y²) dA over the part of the ring 1 ≤ r ≤ 2 with π/4 ≤ θ ≤ 3π/4 (45° to 135°).”

Use this for “A block 2 m by 3 m and 4 m tall has density rising evenly from 0 at the base to 600 kg/m³ at the top. Find its mass and center of mass.”

Use this for “F = ∇φ with φ = 2x² − xy + y². Find the work done by F along any path from (1, 1) to (2, 3).”

Use this for “Find the outward flux of F = ⟨3x, 3y, 3z⟩ through the sphere x² + y² + z² = 4.”

Use this for “A wire runs straight from (1, 0) to (4, 4) with density f(x, y) = x + 2y. Find its mass ∫ f ds.”

Use this for “Use the Divergence theorem to find the flux of F = ⟨x², y², z²⟩ out of the box 0 ≤ x ≤ 2, 0 ≤ y ≤ 1, 0 ≤ z ≤ 3.”

Use this for “Use Stokes’ theorem to find the circulation of F = ⟨−2y, 2x, 0⟩ round the circle x² + y² = 4, z = 0.”

Use this for “Which theorem turns the flux of F out of a closed surface into a triple integral?”

Use this for “Find the curvature and the radius of curvature of y = x² at x = 1.”

Use this for “Find the volume of the solid inside the sphere ρ = 2 and the cone φ = π/3 (60°).”

Use this for “With x = 2u + v, y = u + 3v, find the area of the image of the rectangle 0 ≤ u ≤ 1, 0 ≤ v ≤ 2.”