Let F(x, Y, Z)be a Function of Three Variables $\int _ { C } \operatorname { grad } f \cdot \overline { d r }$

question 68

Short Answer

Let f(x, y, z)be a function of three variables.Suppose that C is an oriented curve lying on the level surface f(x, y, z)= 2.
Find the line integral $\int _ { C } \operatorname { grad } f \cdot \overline { d r }$

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Definitions:

High Tariffs

Taxes imposed on imported goods to protect domestic industries from foreign competition by making imported goods more expensive.

Second Industrial Revolution

A period of rapid industrial growth and technological advancement during the late 19th and early 20th centuries, often associated with new innovations in steel production, electricity, and transportation.

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The extraction of valuable minerals or other geological materials from the earth.

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A sector centered around the development, maintenance, and operation of railways, which played a critical role in economic growth and the expansion of trade and transport in the 19th and early 20th centuries.