Let ${ \vec { F } } = a \vec { i } + b \vec { j } + c \vec { k }$

question 7

Essay

Let ${ \vec { F } } = a \vec { i } + b \vec { j } + c \vec { k }$ be a constant vector field and S be an oriented surface.
Show that $\int _ { S } \vec { F } \cdot \vec { d A } \leq \left( \sqrt { a ^ { 2 } + b ^ { 2 } + c ^ { 2 } } \right) ( \text { Area of } S )$

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Definitions:

Producer Surplus

The difference between what producers are willing and able to supply a good for and the price they actually receive, measuring the benefit to producers.

Total Surplus

The total net gain to society from creating and consuming a product or service, encompassed by the combination of consumer and producer surplus.

Deadweight Loss

An inefficiency in the economy that results when the market equilibrium for a good or service isn't attained or is unattainable.

Tariff

A tax imposed on imported or, less commonly, exported goods with the aim of raising the cost to influence trade dynamics.