Math 6B
Final Exam Review Sheet
March 12, 2025
1. Let g(x1, x2, x3) = x2 − x1 and A be that part of the cone x12 = x22 + x32 with x1 ≤ 1. Evaluate the surface integral g dS.
2. Show that the tangent plane to the cone x32 = x12 + x22 at every point on the cone contains the origin.
3. Give a parameterization of the following surfaces.
(a) The section of z − 3y + x = 2 inside the cylinder x2 + y2 = 4.
(b) x2 + y2 − z2 = 4.
(c) The section of x2 + y2 = z2 in the third octant.
(d) ellipsoid:
4. Let A be the surface z = 10 − x
2 − 2y
2
. Give the equation of the tangent plane to A at (1, −2, 1) in the following ways:
(a) By using the parameterization r(u, v).
(b) By viewing S as a two-variable function g(x, y).
(c) By viewing S as the level surface of a three-variable function g(x, y, z).
5. Consider the vector field Let C = C1 ∪ C2 ∪ C3 ∪ C4 be a positively oriented simple closed curve (see the graph below), where C1, C3 are line segments, and C2, C4 are quarters of circles centered at the origin with radii 2 and 1, respectively.
(a) Use Green’s theorem to compute R · dr, the line integral of R(x, y) over C.
(b) Without any calculations, determine the value of the line integral based on how the vector field R(x, y) looks like at each point. Justify your answer.
6. Evaluate the flux of R(x, y, z) = xˆi + y
ˆj + z kˆ out of a closed cylinder with radius 2 centered on the x-axis with −3 ≤ x ≤ 3.
7. Evaluate the surface integral G · dS, where G = x
2yzˆi + xy2
z
ˆj + xyz2 kˆ, and S is the surface of a cube with faces x = 0, x = 1, y = 0, y = 1, z = 0, z = 1,
(a) directly,
(b) using Divergence theorem.
8. Let R = r|r|, where r = xˆi + yˆj + zˆk. Let S be the surface of the hemisphere along with the disc x
2 + y
2 ≤ 1. Evaluate R · dS.
9. Evaluate the line integral
where
(a) C is the circle x
2 + y
2 = 1. Why aren’t we allowed to use Green’s theorem in this case?
(b) C is the circle (x − 1)2 + (y − 1)2 = 1.
10. Evaluate the work done by the force field F(x, y) = ⟨x, x2 + 3y
2
⟩ on an object moving along the straight line segments (0, 0) → (4, 0) → (2, 4) → (0, 0), which is a triangle.
11. Let R(x1, x2, x3) = 2x1
ˆi + x1x2
2 ˆj + x1x2x3 kˆ and S be the surface boundary of the solid bounded by x1
2 +x2
2 = 1, x1
2 +x2
2 = 4, x3 = 0, and x3 = 4. Evaluate the flux of R out of S.
12. Suppose A is that section of the plane x2 − 2x3 = 2 inside the ellipsoid x12 + 4/x22 + x3 2 = 1, and c is the boundary of A, the intersection curve, oriented clockwise when viewed from the origin. The vector field R(x1, x2, x3) = ⟨x2, x3, x1⟩ is defined on R
3
. Evaluate the line integral H R · dr first directly and then using Stokes’ theorem.
13. Let c consists of the line segments (0, 0, 3) to (4, 0, 3), (4, 0, 3) to (4, 4, 4), (0, 4, 4) to (0, 0, 3), and the curve x2 = 4/x12 − x1 + 4 with x3 = 4 connecting (4, 4, 4) to (0, 4, 4), oriented clockwise when viewed from the origin. Let R(x1, x2, x3) = ⟨x3 − 1, x2 cos (x3π), 0⟩. Evaluate the line integral H R · dr first directly and then using Stokes’ theorem.
14. Let f(x) = x
2 on [−1, 1] be given.
(a) Give a formula for the Fourier series of this function.
(b) Using the Fourier series you obtained in (a), show that
15. Show that the trigonometric identity
could be interpreted as a Fourier series. Use this identity to obtain the Fourier series of cos4 x without finding the Fourier series coefficients directly.
16. Consider the Fourier series of the function f(x) = x on [0, l]. Assume the series could be integrated term by term (this has to be justified since the series is infinite).
(a) What is the Fourier cosine series of 2/x
2 ? What is the constant of integration, which is the first term in the series?
(b) Let x = 0. Evaluate the series
17. Consider the function a(x) = x|x| on (−1, 1) as a periodic function with p = 2. Evaluate the Fourier series of this function.
18. Evaluate even and odd extensions of the function
19. Consider the wave equation
utt = c2 uxx,
where x ∈ [0, l] and with the initial values u(x, 0) = η(x) and ut(x, 0) = δ(x), and homogeneous Dirichlet boundary conditions
u(0, t) = u(l, t) = 0.
Use the method of separation of variables to give a solution to this equation. Use this method to give a solution to the heat equation with the initial value u(x, 0) = η(x) and homogeneous boundary conditions
u(0, t) = u(l, t) = 0.