Vector Calculus - 7

 If V is the volume enclosed by surface S and  =  then is

If C is a smooth curve in R3 from (–1, 0, 1) to (1, 1, –1), then the value of is

  where P is a vector, is equal to

if and  be the set of orthonormal unit vectors, then  is

Stoke’s theorem is

The relation between the line integral and the surface integral is

The divergence of the vector field

If Φ(x, y, z) = 3x²y - y³z²  then the value of grad Φ at the point (1,- 2 ,- 1 ) is

If Φ is a differentiable scalar point function, then the value of curl grad Φ is

If  then  is

A particle moves along the curve Acceleration of the particle in the direction of the motion is

Using stokes' theorem evaluate the line integral, where L is the intersection of x² + y² + z² = 1 and x + y = 0 traversed in the clockwise direction when viewed from the point (1, 1, 0)

Gauss’s divergence theorem can be written a

Let is a solution of the Laplace equation then the vector field is 

Let  be a vector field.
Q. The value of div  is

Let  be a vector field.
Q. The value of curl  is

Let  and 
Q. The value of  is