VCFT: Cat 2

Gauss's theorem of vector calculus connects

• volume integral with surface integral

• surface integral of divergence of a vector with volm integral of a scalar

• surface integral of divergence of a vector with volm integral of the same vector

• surface integral of divergence of a vector with volm integral of a different vector

Surface integral of curl of a vector is connected with its line integral through

• Green's theorem

• Stoke's theorem

• Guass's theorem

• Laplace's theorem

To move from Green's Theorem to Stoke's Theorem we need to change

• from 2 dimensions to 3 dimension

• a line integral to a surface integral

• the double integral of curlF⋅kcurl⁡F⋅k over a region DD in the plane to a surface integral of curlF⋅ncurl⁡F⋅n over a surface floating in space

• the component of the curl to the tangential vector

The surface does not have to be flat inside. True or False

• True

• False

Does the orientation matter?

• Yes

• No

When a vector is irrotational, which condition holds good?

• a)Stoke’s theorem gives non-zero value

• b) Stoke’s theorem gives zero value

• c) Divergence theorem is invalid

• d) Divergence theorem is valid

The Cartesian coordinates can be related to cylindrical coordinates and spherical coordinates. State True/False.

• True

• False

Determine the integral rule of the given expression:

• cosx + c
• -cosx + c
• ½sin²x + C
• ¹/_√(1-x²) + C

Determine the integral rule of the given expression:

• x^-2 + C
• ^-2/_x³ + C
• ^-1/_x + C
• ²/_x + C

Evaluate $\oint_c^{ }\left(7y-e^{\sin x}\right)dx+\left[15x-\sin\left(y^3+8y\right)\right]dy$   where c is the circle with radius 3 centered at (5,-7)

hint: use green's theorem

• 9 $\pi$  

•  $\pi^2$  

•  $72\pi$  

• can't be calculated