Vector Analysis

Vector Analysis
Level 1 Objective Problems
1) Which of the following second derivatives are zero?
a)
b)
c)
d)
e)
2) Which of the following are correct?
a)
b)
c)
d)
3) The 3-dimensional coordinate system is rotated about the x-axis through an
angle θ counter clockwise. Which of the following is a correct
transformation from (x, y, z) to (x', y', z') (T in the superscript stands for
transpose of the row vector)?
a)
b)
c)
d)
Level 2 Objective Problems
1) Tick the correct option:
a)
b)
and
+
c)
+
and
d) None of the above.
2) Consider a generalized coordinate system (u, v, w) for which
Which of the following are correct statements?
a) In cylindrical coordinates, where
b) In spherical coordinates,
and
c) For any scalar function T,
.
d) In spherical coordinates,
3) Which of the following are INCORRECT?
a)
b) A vector field E has the value
inside a unit sphere centred at the origin
and 0 outside. Then,
everywhere.
c)
d) The graph of
is the limit of triangles with base extending from
to
e)
and vertex at
.
at the point
.
Subjective Problems
1) Prove that for a scalar field
where the
surface S has the boundary C.
2) Compute the divergence of the function
Check the divergence theorem for this function, using your volume as the
inverted hemispherical bowl of radius R, resting on the xy-plane and
centred on the origin.
[Use
and
]
3) What is the charge density of a uniform, infinitesimally thin spherical shell
of radius R and total charge Q, centred at the origin?