Find the Value of the Constant $c$ Such That the Vector Field

question 29

Short Answer

Find the value of the constant $c$ such that the vector field
$$\mathbf { G } ( x , y , x ) = \left( 6 x + 2 y ^ { 3 } + 8 z \right) \mathbf { i } + \left( 6 x ^ { 3 } + 2 y + 4 z \right) \mathbf { j } + ( c z - x ) \mathbf { k }$$
is the curl of some vector field $\mathbf { F }$ .

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Definitions:

Ventricles

Chambers of the heart that receive blood from the atria and expel it to the lungs or body.

Isovolumetric Contraction

A phase of the cardiac cycle where the ventricles contract with no change in volume, occurring between the closure of the atrioventricular valves and the opening of the semilunar valves.

Semilunar Valves

Heart valves located at the bases of the large arteries that leave the heart; they prevent blood from flowing back into the heart.

AV Valves

The atrioventricular valves, consisting of the tricuspid and mitral valves, regulate blood flow from the atria to the ventricles in the heart.