Vector Calculus MCQ Quiz - Objective Question with Answer for Vector Calculus - Download Free PDF

Explanation:

v . curl v = v . ∇ × v = [v ∇ v] = 0

Option (4) is true. 

Concept:

The divergence of a vector field is given by:

Given:

, and the cube lies in

Calculation:

Since has no component,

Find partial derivatives at the center of the cube:

At

At

Final Result:

Correct Answer: 4) 64

Explanation:

(a) 

(This is always true for any three given vectors)

(b) We know that  is always true but  because 

This can be true only when 

So, 

This can be true if

 this is true in this case

(c)  can't be equal to  because  and 

So, 

(d) 

0 = 0

(Hence this is proved) 

Explanation:

To evaluate, ∯_s z dx dy

where S(sphere) = x² + y² + z² = R² 

= ∯z dxdy

Using Gauss Divergence Theorem:

= Volume of sphere = 

Explanation:

𝑆 is the portion of the plane 𝑧 = 5𝑥 + 3𝑦 − 100 which lies inside the cylinder 𝑥2 + 𝑦2 = 3.

z_x = 5, z_y = 3

then surface area of S

=  

=  

=  

= 

Now, the cylinder is 𝑥2 + 𝑦2 = 3

then surface area of S

=  = √35 × π ×  = 3√35 π

Given area S = απ

Hence α =   = 17.74

17.74 is the answer.

Explanation

Given

Integral

∫∫∫v 8 xyz dv 

Limits for x, y and z is given as

[2, 3] × [1, 2] × [0, 1]

Volume of the integral

V = ∫∫∫v 8 xyz dv 

i.e. V = ∫ ∫ ∫_V 8 xyz dxdydz

V = 5 × 3 × 1

V = 15 

∴ Volume is 15 

Concept:

If a triangle is formed by three vectors, then the sum of the vectors should be zero.

AB + BC + CA = 0 

Cross product of the vectors:

For two vectors  and  the cross, the product is given by: 

The magnitude of the cross product is:

Area of a triangle:

If the vectors  form adjacent sides of the triangle then the area of the triangle is given by: 

Calculation:

Given:

Let, AB = 3i + 4j and CA = 5i +7j + k

If a triangle is formed by three vectors, then the sum of the vectors should be zero.

AB + BC + CA = 0 ⇒ 3i + 4j  + BC + 5i +7j + k = 0

BC = - 8i - 11j - k

Let the adjacent vectors be AB (a), AC (b)   , and .

First, we will calculate the cross product as follow:

Therefore, the magnitude of the cross product is:

Using the formula for the area of the triangle, the area is given by:

Explanation:

Position vector 

Concept:

The curl of a vector is given by the expansion of the following matrix

If 

Then

Calculation:

Given vector is

Than 

Concept:

When two points (x₁, y₁. z₁) and (x₁, y₁. z₂) are mentioned find the relation in terms of the third variable in terms of x,y, and z:

Put the value of z,y, and z and use the end-points of one variable.

Calculation:

Given:

, A (0, 0, 0) and B(1, 1, 1).

Equation of line i.e. path

Concept:

Revolution about x-axis: The volume of the solid generated by the revolution about the x-axis, of the area bounded by the curve y = f(x), the x-axis and the ordinates x = a and x = b is

similarly for revolution about y-axis:

Calculation:

Given:

Hence the required volume will be .

Concept:

If  are any non-zero vector then

Calculation: 

Given, 

= 0

Concept:

Calculation:

Given:

As  

∴

Hence 

Given that,

F = a_x y - a_y x

y = x²

By differentiating on both sides,

⇒ dy = 2x dx

In the given graph, the limits of x are: -1 to 2.

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