For the following integral, say whether Stokes' Theorem, the Divergence Theorem, or neither applies. $\dot { \mathrm { Q } } ^ { \operatorname { curl } } \left( 2 x \vec { i } + z \vec { j } + 2 y ^ { 3 } \vec { k } \right) \times \vec{ d A }$ where S is a triangular plane in space oriented $\text { dow nw ard. }$
A)Stokes' Theorem
B)Divergence Theorem.
C)Neither

Question

Quizplus · Accepted Answer

Stokes' Theorem applies because the integral involves the curl of a vector field dotted with a differential area element, $\vec{dA}$, which is exactly the scenario where Stokes' Theorem is used to relate a surface integral to a line integral around the boundary of the surface.

For the Following Integral, Say Whether Stokes' Theorem, the Divergence Q˙curl⁡(2xi⃗+zj⃗+2y3k⃗)×dA⃗\dot { \mathrm { Q } } ^ { \operatorname { curl } } \left( 2 x \vec { i } + z \vec { j } + 2 y ^ { 3 } \vec { k } \right) \times \vec{ d A }Q˙​curl(2xi+zj​+2y3k)×dA

For the Following Integral, Say Whether Stokes' Theorem, the Divergence $\dot { \mathrm { Q } } ^ { \operatorname { curl } } \left( 2 x \vec { i } + z \vec { j } + 2 y ^ { 3 } \vec { k } \right) \times \vec{ d A }$