Line Integrals and the Gradient

By now we should be comfortable with the meaning of the fundamental theorem of calculus: adding up all the infintesimal changes of a function over an interval gives the total net change over that interval. And once we broaden our horizons and try to understand the calculus of functions $f:\mathbb{R}^n\to\mathbb{R}$ a good question to ask ourselves is "is there a reasonable generalization of the fundamental theorem of calculus here?" It turns out there is, but it involves a new kind of integral (called a line integral) and a new kind of derivative (called the gradient).

Functions $\mathbb{R}^n\to\mathbb{R}$

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Line Integrals and the Gradient

Functions $\mathbb{R}^n\to\mathbb{R}$

The Line Integral

The Gradient