(A)Compute the Flux of the Vector Field $\vec { F } = y \vec { i } - x \vec { j } + 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) z \vec { k }$

question 32

Essay

(a)Compute the flux of the vector field $\vec { F } = y \vec { i } - x \vec { j } + 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) z \vec { k }$ through S_a, the sphere of radius a, $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ , oriented outward.
(b)Find $\lim _ { a \rightarrow 0 } \int _ { S _ { 0 } } \vec { F } \cdot d \vec { A }$ .

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Definitions:

Control Limits

The boundaries in control charts that distinguish between normal and abnormal variations of the process being monitored.

Quality Control Measures

Quality control measures are the techniques and activities used to ensure that the quality of a product or service meets certain standards and specifications.

Special Causes

Unusual factors or events that impact the process and result in variations not accounted for by the existing process model.

Quality Control Variables

Variables that are monitored and controlled during a manufacturing or service process to ensure the quality of the output.