Use Stoke's Theorem to Evaluate $C$ Is the Curve of Intersection of the Hyperbolic Paraboloid

question 43

Short Answer

Use Stoke's theorem to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r } , \mathbf { F } ( x , y , z ) = 6 y x ^ { 2 } \mathbf { i } + 2 x ^ { 3 } \mathbf { j } + 6 x y \mathbf { k }$ ,
$C$ is the curve of intersection of the hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 25$ oriented counterclockwise as viewed from above.

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Definitions:

Motion Parallax

A depth cue in visual perception where objects closer to the observer move faster across the visual field than objects further away.

Depth

The distance from the nearest to the farthest objects in a scene, giving a three-dimensional perspective.

Cue

A signal or stimulus that guides or prompts an action, response, or decision.

Relative Displacement

A measure of the change in position of an object in relation to a reference point.