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Mathematics Homework: Vector Calculus Problems, Exercises of Abnormal Psychology

A collection of vector calculus problems for homework, covering topics such as line integrals, gradient and divergence of scalar and vector fields, and the relationship between gradient, divergence, and curl. Students are expected to prove various identities and find representations of vectors in different coordinate systems. The problems involve calculating integrals, finding directions of maximum rate of change, and evaluating gradients and curl. Suitable for advanced undergraduate students in mathematics or physics.

Typology: Exercises

2015/2016

Uploaded on 10/17/2016

13176777
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Homework 1
Due: October 6, 2016
Problem 1. Prove V
(ψ2ϕϕ2ψ)dv =I(ψ
ϕϕ
ψ)·ˆnda
Problem 2. Show that
A,
B,
Care not linearly independent if
A·(
B×
C) = 0
Problem 3. For any two scalar functions ϕand ψ, prove that
2(ϕψ) = ϕ2ψ+ψ2ϕ+ 2
ϕ·
ψ
Problem 4. If r is the vector from the origin to (x, y , z) prove that
· r = 3 ,
× r = 0 ,
(u ·
)r =u for any vector u
Problem 5. Derive the metric tensor and scaling factor of cylindrical coordinate
Problem 6. If
A= 2rˆar3rsinϕ ˆaθ, find the representation of
Ain the cartesian coordi-
nate system
Problem 7. A vector c in cartesian coordinate system is c = 3 ˆax+ ˆay3 ˆaz. Find its
representation using the following base vectors
ˆa1=1
2( ˆax+ ˆay),ˆa2=1
2( ˆaxˆay),ˆa3= ˆaz
Problem 8. Evaluate S
D·ds when
D=rsinθ ˆar+rsinθ ˆaθ, and Sis a unit sphere
centered at the origin
Problem 9. For g= 2xy +z2, find
a) the magnitude and direction of the maximum rate of change of gat point P(1,3,2)
b) the rate of change of galong the line directed from P(1,3,2) to P(2,2,1), evaluated
at P(1,3,2)
Problem 10. Find
×
Bif
B= 4rsinθ ˆar+ 3rcosϕ ˆaθ
Problem 11. Given f=rsinθcosϕ calculate
f,
×
fand
2f
Problem 12. Prove
2
A=
(
·
A)
× (
×
A) in cartesian coordinate
1

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Homework 1

Due: October 6, 2016

Problem 1. Prove

V

( ∇^2 ϕ ϕ∇^2 )dv =

I

( ∇ ⃗ ϕ ϕ⃗∇ )  ndaˆ

Problem 2. Show that A;⃗ B;⃗ C⃗ are not linearly independent if A⃗  ( B⃗  C⃗ ) = 0

Problem 3. For any two scalar functions ϕ and , prove that

∇^2 (ϕ ) = ϕ∇^2 + ∇^2 ϕ + 2 ∇⃗ ϕ  ∇⃗

Problem 4. Ifr⃗ is the vector from the origin to (x; y; z) prove that ∇ ⃗ r⃗ = 3 , ∇ ⃗ r⃗ = 0 ,

u(⃗  ∇⃗ r)⃗ =u⃗ for any vectoru⃗

Problem 5. Derive the metric tensor and scaling factor of cylindrical coordinate

Problem 6. If A⃗ = 2r aˆr 3 rsinϕ aˆθ, find the representation of A⃗ in the cartesian coordi- nate system

Problem 7. A vectorc⃗ in cartesian coordinate system isc⃗ = 3 ˆax + ˆay 3 ˆaz. Find its representation using the following base vectors

aˆ 1 =

p 2

( ˆax + ˆay); aˆ 2 =

p 2

( ˆax aˆy); aˆ 3 = ˆaz

Problem 8. Evaluate

S

D  d⃗s when D⃗ = rsin aˆr + rsin aˆθ, and S is a unit sphere centered at the origin

Problem 9. For g = 2xy + z^2 , find

a) the magnitude and direction of the maximum rate of change of g at point P (1; 3 ; 2)

b) the rate of change of g along the line directed from P (1; 3 ; 2) to P ′(2; 2 ; 1), evaluated at P (1; 3 ; 2)

Problem 10. Find ∇ ⃗ B⃗ if B⃗ = 4rsin aˆr + 3rcosϕ aˆθ

Problem 11. Given f = rsincosϕ calculate ∇⃗ f , ∇ ⃗ ∇⃗ f and ∇⃗^2 f

Problem 12. Prove ∇⃗^2 A⃗ = ∇⃗ (∇ ⃗ A⃗ ) ∇ ⃗ (∇ ⃗ A⃗ ) in cartesian coordinate