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:

Developed Countries

Nations with high levels of industrialization, a high standard of living, and developed infrastructure.

Industrialization

The process by which an economy is transformed from primarily agricultural to one based on the manufacturing of goods, involving increased mechanization.

Per Capita GDP

A measure of the economic output of a nation per individual, calculated by dividing the country's gross domestic product by its total population.

Developed Countries

Nations with advanced economies, high standards of living, and well-established infrastructure and government systems.