PHYB21S- 2022 Assignment-1

Due date: Jan 28, by 5:00 PM Q1: A- Show that B- Show that for any vector function and a scalar function the following relations and is satisfied. Verify the first relations for the case C- Verify the following relations

Q2:
1) Evaluate the integral

given that and the path C1 is the parabola
and the line in the plane from (0, 0, 2) → (1, 1, 2).
Briefly comment on the result.

2) Is the line integral , with the vector field independent of
path? If so find the potential that produces the field.

3) Given that

Choose any path between (0,0,1) and (1, , 2) to evaluate the integral Q3: A. If u and v are vectors, use tensor method to prove that 1. 2. B. Consider the two vectors Show that the following identities are satisfied:
(rˆ ⋅∇)rˆ = 0
v T
∇⋅(∇× v) = 0 ∇× (∇T )
v = −2yxˆ − 3zyˆ − zzˆ
∇( f / g) = (g∇f − f∇g) / g2
∇⋅(

A / g) = [g(∇⋅

A)−

A ⋅(∇g)] / g2
∇× (

A / g) = [g(∇×

A)+

A × (∇g)] / g2


F
C
∫ ⋅dr


F = iˆx2y + jˆ(x − z)+ kˆxyz
y = x2 y = x z = 2


F
C
∫ ⋅dr

F = 2xy2iˆ + 2x2yjˆ
φ
π / 4


F
C
∫ ⋅dr
∇× (u × v) = (∇⋅v)u − (∇⋅u)v + (v.∇)u − (u.∇)v
∇× (∇× v) = ∇(∇⋅v)−∇2v
A = xxˆ + 2yyˆ + 3zzˆ and B = 3yxˆ − 2xyˆ
1. 2. 3.
Q4: a) Simplify the following expressions as much as possible b) For the following equations, write down the equivalent in vector or matrix notations, or explain why the equation is invalid;
c) Use tensor notations to write the following quantities


∇⋅(A × B) = B ⋅(∇× A)− A ⋅(∇× B)
∇(A ⋅B) = A × (∇× B)+ B × (∇× A)+ (A ⋅∇)B + (B ⋅∇)A
∇× (A × B) = (B ⋅∇)A − (A.∇)B + A(∇⋅B)− B(∇⋅A)

欢迎咨询51作业君