9—Vector Calculus 1 234whereu=asinhθ. Put this back into the original variables and you have
φ=−Gλ
[
sinh−^1(
L−z
√
x^2 +y^2
)
+ sinh−^1(
L+z
√
x^2 +y^2
)]
(9.58)
The inverse hyperbolic function is a logarithm as in Eq. (1.4), so this can be rearranged and the terms
combined into the logarithm of a function ofx,y, andz, but thesinh−^1 s are easier to work with so
there’s not much point. This is not too complicated a result, and it is far easier to handle than the
vector field you get if you take its gradient. It’s still necessary to analyze it in order to understand it
and to check for errors. See problem9.48.
Exercises1 Prove that the geometric interpretation of a cross product is as an area.
2 Start from a picture ofC~=A~−B~and use the definition and properties of the dot product to derive
the law of cosines. (If this takes you more than about three lines, start over, and no components, just
vectors.)
3 Start from a picture ofC~ =A~−B~ and use the definition and properties of the cross product to
derive the law of sines. (If this takes you more than a few lines, start over.)
4 Show thatA~.B~×C~=A~×B~.C~. Do this by drawing pictures of the three vectors and find the
geometric meaning of each side of the equation, showing that they are the same (including sign).
5 (a) If the dot product of a given vectorF~ with every vector results in zero, show thatF~ = 0.
(b) Same for the cross product.
6 From the definition of the dot product, and in two dimensions, draw pictures to interpretA~.(B~+C~)
and from there prove the distributive law:A~.=A~.B~+A~.C~. (DrawB~+C~tip-to-tail.)
7 For a sphere, from the definition of the integral, what is
∮
dA~? What is
∮
dA?
8 What is the divergence ofxxyˆ +yyzˆ +ˆzzx?
9 What is the divergence ofˆrrsinθ+ˆθr 0 sinθcosφ+φrˆ cosφ? (spherical)
10 What is the divergence ofˆrrsinφ+φzˆ sinφ+zzrˆ? (cylindrical)
11 In cylindrical coordinates draw a picture of the vector field~v=φrˆ^2 (2+cosφ)(forz= 0). Compute
the divergence of~vand indicate in a second sketch what it looks like. Draw field lines for~v.
12 What is the curl of the vector field in the preceding exercise (and indicate in a sketch what it’s like).