Vector Calculus - 8

# Vector Calculus - 8

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## 20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Vector Calculus - 8

Vector Calculus - 8 for Mathematics 2023 is part of Topic-wise Tests & Solved Examples for IIT JAM Mathematics preparation. The Vector Calculus - 8 questions and answers have been prepared according to the Mathematics exam syllabus.The Vector Calculus - 8 MCQs are made for Mathematics 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Vector Calculus - 8 below.
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Vector Calculus - 8 - Question 1

### If then div curl is equal to

Vector Calculus - 8 - Question 2

### If Φ is a differentiable scalar function, then div grad Φ is equal to

Vector Calculus - 8 - Question 3

### If then is

Detailed Solution for Vector Calculus - 8 - Question 3

Explanation : curl F = ∇ × F

= (∂/∂x, ∂/∂y, ∂/∂z) × (F1, F2, F3) (∂F3/∂y − ∂F2/∂z , ∂F1/∂z − ∂F3/∂x , ∂F2/∂x − ∂F1/∂y)

= (∂z/∂y − ∂y/∂z , ∂x/∂z − ∂z/∂x, ∂y/∂x − ∂x/∂y)

= (0, 0, 0)

Note that since this curl is 0, the radial vector field

F(x, y, z) = (x, y, z) is irrotational.

Vector Calculus - 8 - Question 4

The value of along the circle x2 + y2 = 1 is

Vector Calculus - 8 - Question 5

The value of is

Vector Calculus - 8 - Question 6

Directional derivative of ψ(x,y,z) = xy2 + 4xyz + z2 at the point (1, 2, 3) in the direction of is

Vector Calculus - 8 - Question 7

If and are two vectors then the value of will be

Vector Calculus - 8 - Question 8

The value of div grad Φ is

Vector Calculus - 8 - Question 9

The value of is

Vector Calculus - 8 - Question 10

Gauss’s divergence theorem is

Vector Calculus - 8 - Question 11

If then the value of div will be

Vector Calculus - 8 - Question 12

Let W be the region bounded by the planes x = 0, y = 0, y = 3, z = 0 and x + 2z = 6. Let S be the boundary of this region. Using gauss’ divergence theorem, evaluate, where and ň is the outward unit normal vector to S.

Detailed Solution for Vector Calculus - 8 - Question 12

By gauss divergence theorem    Vector Calculus - 8 - Question 13

If is a differentiable vector point function, then the value of div curl is

Vector Calculus - 8 - Question 14

Let and Q. The unit vector perpendicular to the plane containing and is

Detailed Solution for Vector Calculus - 8 - Question 14

Given that and Therefore, The unit vector perpendicular to the plane containing vector and is  Vector Calculus - 8 - Question 15

Let us consider the scalar point function f(x y, z) =x2 + y2 + z2
Q. The grad of f(x, y, z) is

Detailed Solution for Vector Calculus - 8 - Question 15

Grad f=  Vector Calculus - 8 - Question 16

Let us consider the scalar point function f(x y, z) =x2 + y2 + z2
Q. The directional derivative of f(x, y, z) at the point P(1, 1, 1) along is

Detailed Solution for Vector Calculus - 8 - Question 16

(Grad f)(1,1,1) Now, the directional derivative o f f at P(1,1,1) along is Vector Calculus - 8 - Question 17

Let where a, b and c are constants and S is the surface of unit sphere.
Q. The value of is

Detailed Solution for Vector Calculus - 8 - Question 17

By Gauss divergence theorem,  Vector Calculus - 8 - Question 18

Let where a, b and c are constants and S is the surface of unit sphere.
Q. The value of is

Detailed Solution for Vector Calculus - 8 - Question 18

By Gauss divergence theorem,  Since, V is enclosed by a sphere of unit radius. Thereofore
a + b + c = 1
and Vector Calculus - 8 - Question 19

A vector normal to is

Detailed Solution for Vector Calculus - 8 - Question 19 We take,  = 1 - 2 + 1 = 0
So, B is normal to A.

Vector Calculus - 8 - Question 20

If and curve C is the arc of the curve y = x3 from (0,0) to (2,8), then the value of Detailed Solution for Vector Calculus - 8 - Question 20

Since, C is the curve y = x3 from (0,0) to (2,8)
So, let x = t ⇒ y = t3
If is the position vector of any point on C, then or or At (0, 0) ⇒ t = x = 0 and at (2, 8) ⇒ t = 2
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