Class Test I, MTH201
DATE: October 13th, 2023
Q 1. (20 Marks) Does the following function
f(x, y, z)
def = xyz for all x, y and z (1)
describe a scalar field or a vector field ? If it is a scalar field, find its gradient. If it is a vector field, find its divergence.
Q 2. (20 Marks) Figure 1 shows a vector field F and a directed curve Γ in the xy-plane. What is the sign of the line integral of F along Γ as indicated in the figure. Justify your answer as well. Answer without any justifications will get only few marks.
Figure 1: A vector field F and a directed curve Γ . All the vectors of F point in the positive direction of y-axis.
Q 3. (20 Marks) Evaluate the line integral of
F(x, y, z)
def = zi + xj + yk for any x, y and z, (5)
along a straight line Γ from (1, 1, 0) to (0, 0, 1) .
Q 4. (20 Marks) Let E be the electric field generated by a postive point charge q that is fixed at the origin O of a coordinate system. Evaluate the surface integral of E through the surface
S
def = {(x, y, z)| − 1 ≤ x ≤ 1, −1 ≤ z ≤ 1, y = 1} (14)
in the positive direction of y-axis (Figure 2) .
Figure 2
Q 5. (20 Marks) Is it possible to find a scalar field u such that its gradient is
▽u = —6xyi + (3y2 — 3x2 )k, for any x,y and z ? (17)
If so, give your example of u and justify your choice. If not, give your reasoning.