Use Stokes' Theorem to Evaluate $\iint _ { S } \operatorname { cur } \mathbf { F } \cdot d \mathbf { S }$

question 7

Short Answer

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { cur } \mathbf { F } \cdot d \mathbf { S }$ .
$$\mathbf { F } ( x , y , z ) = 9 x y \mathbf { i } + 6 y z \mathbf { j } + 9 z ^ { 2 } \mathbf { k }$$ $S$ is the part of the ellipsoid $4 x ^ { 2 } + 4 y ^ { 2 } + 49 z ^ { 2 } = 196$ lying above the $x y$ -plane and oriented with normal pointing upward.

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Definitions:

Cross-sectional Area

The area of a particular section of an object or material that is exposed when it is cut through by a plane.

Deposition

The geological process by which sediments, soil, and rocks are added to a landform or landmass, often carried by wind, water, or ice.

Cutbank

A steep cut or slope formed by lateral erosion of a stream, especially on the outside bend of a channel.

Gradient

The rate of change of a quantity (e.g., temperature, pressure, or elevation) with respect to distance in a given direction.