Calculus Answers

[SADT3] For scalar functions u and v, show that

B=( nabla u)*( nabla v)

is solenoidal and that

A= 1 2 (u nabla v-v nabla u)

is a vector potential for B, i.e. B= nabla* A

The Laplacian of a function f

 of n

 variables x

1

,x

2

,⋯x

n

, denoted ∇

2

f

 is defined by

∇

2

f(x

1

,x

2

,⋯,x

n

):=∂

2

f

∂x

2

1

+∂

2

f

∂x

2

2

+⋯+∂

2

f

∂x

2

n

Now assume that f

 depends only on r

 where r=(x

2

1

+x

2

2

+⋯+x

2

n

)

1

2

, i.e. f(x

1

,x

2

,⋯,x

n

)=g(r)

, for some function g

. Show that, for x

1

,x

2

,⋯,x

n

≠0

,

∇

2

f=n−1

r

g

′

(r)+g

′′

(r)

 If A

 and B

 are vector fields, prove the following:

∇(A⋅B)=(B⋅∇)A+(A⋅∇)B+B×(∇×A)+A×(∇×B).

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1. D²((x+1)²/x-1)            ans, 8/(x-1)³
2. D(ln(x²-2x+1))            ans, 2/x-1

1. D(x²+2x-3)                  ans, 2x+2
2. D²(xe^3x-e^4x)                 ans, 9xe^3x+63^3x-16e^4x

Evaluate the following functions in differential operator form.

1. D2((x+1)2/x-1) Answer: 8/(x-1)³
2. D(ln(x²+2x+1) Answer: 2/x-1

A manufacturer packages his product in a 500-ml cylindrical can. The material for the

top costs four times as much per square centimeter as that to be used for the bottom and

the sides. What must be its dimensions so that the cost of the can is least?

Find the extreme values of z

 on the surface 2x

2

+3y

2

+z

2

−12xy+4xz=35