Vector Calculus MCQ Level - 1 - Physics MCQ

The angle between the surfaces at point is the angle between the normals to the surfaces at the point.

A normal to z = x² + y² - 3 at (2, -1, 2) is

A normal to x² + y² + z² = 9 at (2, -1, 2) is

Let angle between surfaces ϕ₁ and ϕ₂ is θ.

By Stoke’s theorem,

∴  we need to evaluate 


= –4
The correct answer is: –4

Now y = x³
Differentiating both sides,
dy = 3x² dx
Now, r = x Î + y Ĵ
Differentiating both sides,
Now, dr = dx Î + dy Ĵ
Now F.dr = (5xy - 6x²)dx + (2y - 4x) dy
Substituting dy = 3x² dx and y = x³ in the above equation

The correct answer is: 35

Normal to the surface  = constant will be :

The correct answer is: 81

By the definition: A vector field whose divergence comes out to be zero or Vanishes is called as a Solenoidal Vector Field. i.e. 

If is a Solenoidal Vector field.

Correct Answer :- c

By Green’s Theorem,

Explanation : ∫3x²e^y dx + ey dy

= ∫∫-3x²e^y dxdy

= -3*4 ∫(1 to 2) x² dx ∫(1 to 2) e^y dy 

= 12[x³/3](1 to 2) [e^y](1 to 2)

= -4(8-1)(e²-e)

= -28(e²-e)

By Gauss Divergence Theorem,

The correct answer is: 30

which is given to be 
Hence,  

for b = 2 and a being any value.

The correct answer is: (1, 2)

The correct answer is:  

Also, the normal to the surface z – x = constant is



The correct answer is: -48π