Multivariable calculus problems
This is the fourth midterm exam for Math 210. It consists of 15 questions, for a total of 75 points. You will have 75 minutes to complete it. Show all your work in the provided white-space under each problem and clearly indicate your answer. Partial credit may be given generously, even for wrong answers, if you demonstrate understanding of what your instructor has taught you. A wrong answer with no work shown earns zero points. A right answer with no work shown is suspicious. By taking this exam you agree to respect Chapman University’s policies concerning academic honesty. In particular, you will not copy from others during this exam and will not communicate exam topics with anyone until all exams are collected. You may only use a basic scientific calculator, no cell phones or computers, or notes of any kind are permitted. Please read each question carefully and follow instructions. If you finish early, double check all of your solutions for completeness and correctness. Page: 1 2 3 4 5 6 7 Total Points: 10 15 10 10 10 10 10 75 Score: This page has been left blank. Feel free to use it as scratch paper. 1. (5 points) Compute∫ γ xyds where γ is the straight line segment connecting (1, 1) to (2, 3). 2. (5 points) Compute∫ γ F⃗ · T⃗ ds where F⃗ = ⟨−y, x, z⟩ and γ is the helix γ = ⟨cos(t), sin(t), t⟩ 0 ≤ t ≤ π 3. (5 points) Show that the vector field F⃗ = ⟨−y2, x2⟩ is not conservative. 4. (5 points) The vector field F⃗ = ⟨yz + 2xz2 + y, 2yz + x+ xz, y2 + 2x2z + xy⟩ is conservative. Find a function f such that ∇f = F⃗ 5. (5 points) For the vector field F⃗ = x⃗i+ yj⃗√ x2 + y2 compute∫ γ F⃗ · T⃗ ds along the curve x = (1 + t π ) cos(t2 + (1− 2π)t) y = 2t π + sin(t) 0 ≤ t ≤ 2π Work smarter, not harder. This is easy if you approach it with the right theorem. 6. (5 points) Find the area of the region enclosed by the curve x = t(t− 2)et y = t2(t− 2)et 0 ≤ t ≤ 2 7. (5 points) Consider the annulus Ω of all points 1 ≤ x2 + y2 ≤ 4 and the vector field F⃗ = ⟨x sin (π 6 ( x2 + y2 − 1 )) , y sin (π 6 ( x2 + y2 − 1 )) ⟩ Compute∫∫ Ω div(F⃗ )dA Again: work smarter, not harder. 8. (5 points) Find the surface area of the helicoid ⟨u cos(v), u sin(v), v⟩ 0 ≤ u ≤ 1, 0 ≤ v ≤ 2π 9. (5 points) Let Σ be the helicoid as in the previous question. Compute the surface integral∫∫ Σ √ x2 + y2dA 10. (5 points) Let Σ be the part of the paraboloid z = 9− x2 − y2 where z ≥ 0. Compute the flux integral across Σ of F⃗ = ⟨x, y, z⟩, using the upward pointing unit normal to Σ. 11. (5 points) An easier problem: Compute the curl of the vector field F⃗ = ⟨x2 + yz, y2 + xz, z2 + xy⟩. Is F⃗ conservative? Stokes’ Theorem Stuff 12. (5 points) Suppose Σ is an oriented surface in space with two disjoint boundary curves γ1 and γ2 which are oriented positively with respect to Σ. If F⃗ is a vector field where∫ γ1 F⃗ · Tds = 2 and ∫ γ2 F⃗ · Tds = 5, what is the value of∫∫ Σ curl(F⃗ ) · ndA? 13. (5 points) In electromagnetism, if E⃗ is a constant in time electric field and B⃗ is the magnetic field, then curl(B⃗) = 4π c J⃗ where c is some constant (the speed of light) and J⃗ is the current density field, meaning the current passing through a surface is the flux integral∫∫ Σ J⃗ · dA⃗ Supposing that the magnetic field is conservative in a region Ω where the electric field is constant in time, how much current passes through any surface in the region? Divergence Theorem Stuff 14. (5 points) Use the divergence theorem to compute the integral∫∫∫ Ω x2dV where Ω is the inside of the torus x = cos(u) y = (sin(u) + 2) cos(v) z = (sin(u) + 2) sin(v) 0 ≤ u ≤ 2π, 0 ≤ v ≤ 2π 15. (5 points) Compute the flux integral of F⃗ = ⟨yz, x sin(z2), x2⟩ across the hemisphere x2 + y2 + z2 = 1, z ≥ 0, with upward pointing unit normal. Clever use of the divergence theorem can save you a lot of work. Think about capping off this hemisphere.