Quiz 1 Oct. 20th 2022
Q 1. (20 Marks) Find a vector field F such that ▽ × F = 0 at the point (1, 1, 1), and ▽ · F = 1 at the point (0, 0, 0) . Show your reasoning and/or computation for your answer. If you think that the F that satisfies both requirements does not exist, then explain with your reasons.
Q 2. (20 Marks) Suppose F = yi + zj + xk. Let Γ be a directed straight line from (1, 0, 1) to (0, 1, 0) . The T is the orientation of Γ . Evaluate the line integral ,Γ F · Tds.
Q 3. (20 Marks) Let Γ be a circle in the xy-plane. The Γ is centered at the origin and has radius 1 . The orientation T of Γ is anti-clockwise. Find a vector field F such that
1 < fΓ F · Tds < 2
Show your reasoning and/or computation for your answer.
Q 4. (20 Marks) . Let S1 = {(x,y, z)jx2 + y2 + z2 = 1, z ≥ 0}. S2 = {(x,y, z)jx2 + y2 ≤ 1, z = 0}. The n1 is unit normal vector to S1 and points away from the origin. The n2 is the unit normal vector to S2 and points in the positive direction of z-axis. Find a vector field F such that
Z
S1 F · n1 dS >Z
S2 F · n2 dS (1)
Show your reasoning and/or computation for your answer.
Q 5. (20 Marks) Let Ω be a cube centered at the origin and with sides of length 2. In other words, Ω = {(x,y, z); —1 ≤ x ≤ 1, —1 ≤ y ≤ 1, —1 ≤ z ≤ 1}. Let S be the boundary surface of Ω . Let n be the outward unit normal vector of S. Find a vector field F such that gS F · ndS = π . Show your reasoning and/or computation for your answer.