Multivariable Calculus · free preview

Conservative Fields and Potentials

Path independence as the defining property of conservative fields, the fundamental theorem for line integrals, and recovering potentials from gradient fields.

1. The class of fields where paths stop mattering

The class of fields where paths stop mattering

2. A worked potential computation

A worked potential computation

3. Recovering the potential

Recovering the potential

4. Why gradients deserve the name

Why gradients deserve the name

5. The fundamental theorem application

Using $V = x^2 + y^2$ , what is the work of $\nabla V$ from $(1, 0)$ to $(0, 1)$ along any path?

6. Rebuilding a potential

For $\mathbf{F} = (2x y,\ x^2)$ , integrating the first component in $x$ gives $V = x^2 y + g(y)$ . What comes next?

7. The failure that proves non-conservativeness

Rebuilding a potential for $(-y, x)$ leaves $\partial_y(-xy + g(y)) = -x + g'(y)$ against the needed $x$ . What does this prove?

8. Potential of a separated field

For $\mathbf{F} = (2x,\ 2y)$ , give the potential $V$ with zero constant, in the form a^2+b^2.

9. Work as a potential difference

A field has potential $V = \sin(xy)$ , and work is computed from $(1, 0)$ to $(1, \pi/2)$ . Give the work, as a plain number.

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