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

question 5

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:

K

The chemical symbol for potassium, a soft, silvery-white metallic element, essential in the human diet and various industrial applications.

K

The chemical symbol for potassium, an essential mineral and electrolyte involved in nerve function and fluid balance.

[Cl-]

The chloride ion, a chlorine atom that has gained one electron to achieve a stable electronic configuration, commonly found in salt compounds.

HAc

A common abbreviation for acetic acid, a weak organic acid with the formula CH3COOH, known for its sour taste and pungent smell.