Vector Calculus MCQ Quiz in मराठी - Objective Question with Answer for Vector Calculus - मोफत PDF डाउनलोड करा

Concept:

∫_C F(r) dr can be solved by putting:

x = r cos θ

y = r sin θ

dx = -r sin θ dθ

dy = r cos θ dθ

Application:

Given r = 1

θ = 0 to 45°

The required integral can be written as:

With r = 1, the above integral becomes:

r = 1

By stokes theorem,

Here F = (x + y) I + (2x - z) J + (y + z) K

Equation of the plane through A, B, C is

Vector N normal to this plane is

∇(3x + 2y + z - 6) = 3I + 2J + K

 where S is the triangle ABC

Green's theorem:

Stokes theorem:

 = 

Gauss Divergence theorem:

Concept:

The divergence of the given vector field,

Calculation:

Given:

Using equation (1),

Explanation:

If  is twice continuously differentiable, then its second derivatives are independent of the order in which the derivatives are applied. All the terms cancel in the expression for curl , and we conclude that curl⁡∇f=0." id="MathJax-Element-12-Frame" role="presentation" style=" word-spacing: 0px; position: relative;" tabindex="0">

• curl⁡∇f=0." role="presentation" style=" word-spacing: 0px; position: relative;" tabindex="0">Divergence operates on a vector field but results in a scalar.
• Curl operates on a vector field and results in a vector field.
• Gradient operates on a scalar but results in a vector field.
• Divergence of curl, Curl of the gradient is always zero.
• Thus, the gradient of curl gives the result of curl (which is a vector field) to the gradient to operate upon, which is a mathematically invalid expression.curl⁡∇f=0." role="presentation" style=" word-spacing: 0px; position: relative;" tabindex="0">..curl∇f=0.

Straight line from (0, 0) to (2, 4)

x = 2t, y = 4t

dx = 2dt

dy = 4dt

Since x (2t) is from 0 to 2 and y(4t) is from 0 to 4, the limits of t are: 0 to 1

Now, the line integral becomes:

Concept:

In 3D, rotations can also be defined as linear transformations, a rotation in 3D can be represented by a matrix equation P'=RP where R is a rotation matrix.

R =  

where θ is the angle of rotation with the positive axis in the clockwise direction.

Given Data and Calculation:

Given that, Coordinate of point P = (1,1,1)

The angle of rotation = 45°

New Coordinate will be,

= 

So the solution will be 1.3×3" id="MathJax-Element-163-Frame" role="presentation" style=" position: relative;" tabindex="0">rotation matrix.(16)p′=Rp" id="MathJax-Element-161-Frame" role="presentation" style=" position: relative;" tabindex="0">p′=Rp

Concept:

If aî + bĵ + ck̂ and pî + qĵ + rk̂ are 2 different sides of the rhombus.

Suppose  and B = 

Then anyone diagonal of the rhombus is given by 

The other diagonal is given by 

The magnitude of the vector  

Calculation:

Given:

2î + 4ĵ - 5k̂ and î + 2ĵ + 3k̂ are 2 different sides of rhombus.

Suppose,  and B = 

Then anyone diagonal of the rhombus is given by 

D₁ = (2î + 4ĵ - 5k̂) + (1î + 2ĵ + 3k̂)

D₁ = 3î + 6ĵ - 2k̂

The magnitude of the vector diagonal D₁

D₁ = 7 

The other diagonal is given by 

D2 = (1î + 2ĵ + 3k̂) - (2î + 4ĵ - 5k̂)

D2 = - 1î - 2ĵ + 8k̂

The magnitude of the vector diagonal D₂

∴ The length of diagonals of a rhombus is 7 and .

Concept:

Directional derivative of a function f along the vector  is given by:

where 

grad f or ∇ f is defined by the equation,

Calculation:

Given:

z = y²e^2x

At (2, -1), 

Unit vector 

Directional derivative  = 2e4α – 2e4β

Given that, the directional derivative is zero.

⇒ 2e⁴α – 2e⁴β = 0 ⇒ α = β

As b is a unit vector, 

Given vector  

Bounded by open surface of hemisphere x² + y² + z² = 1

Such that z ≥ 0, and closed curve (x² + y² = 1)

We have to find

⇒ from stokes theorem, we have

Now, changing the integral with substitution

x = cos θ ⇒ dx = -sin θ dθ

y = sin θ ⇒ dy = cos θ dθ