Evaluate the Surface Integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$

question 68

Short Answer

Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ for the vector field $\mathbf { F } ( x , y , z ) = - y \mathbf { i } + x \mathbf { j } + 3 z \mathbf { k }$ , where S is the hemisphere $z = \sqrt { 16 - x ^ { 2 } - y ^ { 2 } }$ with upward orientation.

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Definitions:

Variance

A measure of the dispersion or spread of a set of data points around their mean value; it quantifies how much the data points differ from the mean.

Standard Deviation

An indicator of the range or spread of a collection of numbers, showing how far the values diverge from the average.

Observations

The collected data or individual points that have been observed or recorded for a particular study or analysis.

Mean

The mean, or average, of a series of numbers is obtained by dividing the sum of the numbers by how many numbers there are.