Mathematical Tools for Physics

9—Vector Calculus 1 274

Problems

9.1 Use the same geometry as that following Eq. ( 3 ), and take the velocity function to be~v=xvˆ 0 xy/b^2. Take
the bottom edge of the plane to be at(x,y) = (0,0)and calculate the flow rate. Should the result be independent
of the angleφ? Sketch the flow to understand this point. Does the result check for any special, simple value of
φ?

9.2 Repeat the preceding problem using the cylindrical surface of Eq. ( 4 ), but place the bottom point of the
cylinder at coordinate(x,y) = (x 0 ,0). Ans:(v 0 a/4)(2x 0 +πb/4)

9.3 Use the same velocity function~v=ˆxv 0 xy/b^2 and evaluate the flow integral outward from theclosedsurface
of the rectangular box,(c < x < d),(0< y < b),(0< z < a). The convention is that the unit normal vector
points outward from the six faces of the box. Ans:v 0 a(d−c)/ 2

9.4 Work out the details of the divergence of a vector field in spherical coordinates, Eq. ( 16 ).

9.5 For the vector field~v =A~r, that is pointing away from the origin with a magnitude proportional to the
distance from the origin, express this in rectangular components and compute its divergence.
Repeat this in cylindrical coordinates.
Repeat this in spherical coordinates, Eq. ( 16 ).

9.6 Gauss’s law for electromagnetism says

∮

E~.dA~=qencl/ 0. If the electric field is given to beE~=A~r, what

is the surface integral ofE~ over the whole closed surface of the cube that spans the region from the origin to
(x,y,z) = (a,a,a)? What is the charge enclosed in the cube?

(b) What is the volume integral,

∫

d^3 r∇.E~inside the same cube?

9.7 Evaluate the surface integral of~v=ˆrAr^2 sin^2 θ+ˆθBrcosθsinφover the surface of the sphere centered at
the origin and of radiusR. Recall section8.8.