For the following integral, say whether Stokes' Theorem, the Divergence Theorem, or neither applies. $\hat { Q } \left( - 5 x \vec { i } + z \vec { j } + 4 y ^ { 3 } \vec { k } \right) \times \vec { d A }$ where S is the sphere of radius $4$ centered at the origin oriented $\text { outw and }$
A)Stokes' Theorem
B)Divergence Theorem.
C)Neither

Question

Quizplus · Accepted Answer

The Divergence Theorem applies because the integral involves a vector field dotted with a differential area element over a closed surface, which in this case is a sphere. Stokes' Theorem is not applicable because it relates a surface integral over a vector field to a line integral over the boundary of the surface, and in this case, we are dealing with a closed surface (a sphere) without a boundary.

For the Following Integral, Say Whether Stokes' Theorem, the Divergence Q^(−5xi⃗+zj⃗+4y3k⃗)×dA⃗\hat { Q } \left( - 5 x \vec { i } + z \vec { j } + 4 y ^ { 3 } \vec { k } \right) \times \vec { d A }Q^​(−5xi+zj​+4y3k)×dA

For the Following Integral, Say Whether Stokes' Theorem, the Divergence $\hat { Q } \left( - 5 x \vec { i } + z \vec { j } + 4 y ^ { 3 } \vec { k } \right) \times \vec { d A }$