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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
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Problem 1. Prove
V
( ∇^2 ϕ ϕ∇^2 )dv =
( ∇ ⃗ 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