Page 187 - Intro to Tensor Calculus

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EXERCISE 2.1

I 1. In cylindrical coordinates (r, θ, z)with f = f(r, θ, z) find the gradient of f.
~
~
~
I 2. In cylindrical coordinates (r, θ, z)with A = A(r, θ, z) find div A.
~
~
~
I 3. In cylindrical coordinates (r, θ, z)for A = A(r, θ, z) find curl A.
2
I 4. In cylindrical coordinates (r, θ, z)for f = f(r, θ, z) find ∇ f.
I 5. In spherical coordinates (ρ, θ, φ)with f = f(ρ, θ, φ) find the gradient of f.
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~
~
I 6. In spherical coordinates (ρ, θ, φ)with A = A(ρ, θ, φ) find div A.
~
~
~
I 7. In spherical coordinates (ρ, θ, φ)for A = A(ρ, θ, φ) find curl A.
2
I 8. In spherical coordinates (ρ, θ, φ)for f = f(ρ, θ, φ) find ∇ f.
r
I 9. Let ~ = x ˆ e 1 +y ˆ e 2 +z ˆ e 3 denote the position vector of a variable point (x, y, z) in Cartesian coordinates.
r
Let r = |~| denote the distance of this point from the origin. Find in terms of ~ and r:
r
1
m
(a) grad (r) (b) grad(r ) (c) grad( ) (d) grad (ln r) (e) grad(φ)
r
where φ = φ(r) is an arbitrary function of r.
r
I 10. Let ~ = x ˆ e 1 +y ˆ e 2 +z ˆ e 3 denote the position vector of a variable point (x, y, z) in Cartesian coordinates.
Let r = |~| denote the distance of this point from the origin. Find:
r
m
r
r
(a) div (~)(b) div (r ~r)(c) div (r −3 r ~)(d) div (φ~)
where φ = φ(r) is an arbitrary function or r.

r
I 11. Let ~ = x ˆ e 1 + y ˆ e 2 + z ˆ e 3 denote the position vector of a variable point (x, y, z) in Cartesian
coordinates. Let r = |~| denote the distance of this point from the origin. Find: (a) curl ~ (b)curl (φ~)
r
r
r
where φ = φ(r) is an arbitrary function of r.
~
I 12. Expand and simplify the representation for curl (curl A).
I 13. Show that the curl of the gradient is zero in generalized coordinates.
1
2
3
I 14. Write out the physical components associated with the gradient of φ = φ(x ,x ,x ).
I 15. Show that
1 ∂ √ im i 1 ∂ √ i
im
g A i,m = √ gg A m = A = √ gA .
,i
g ∂x i g ∂x i