# Vector field

the vector field $F(x,y)= [ -2 \frac{x}{(x^{2} +y^{2}+1)^2 },-2\frac{y}{(x^{2}+y^{2}+1)^2 } ] $ is conservative throughout the planet. Determine a potential U to F and calculate $\int_{\gamma} \mathbf F \cdot d\,\mathbf r$ where ? is an arbitrary curve in the plane from (3,?4) to (4,1).

calculate U and $\int_{\gamma} \mathbf F \cdot d\,\mathbf r$

## Answer

A potential is:

$$U(x,y) = \frac1{x^2+y^2+1}$$

as you may see via the chain rule:

$$\partial_x U(x,y) = -2\frac x{(x^2+y^2+1)^2},\qquad \partial_y U(x,y) = -2\frac y{(x^2+y^2+1)^2}$$

So for any curve $\gamma$ from $(3,-4)$ to $(1,4)$ we have:

$$\int_{\gamma} F\cdot d\gamma = \int_\gamma \mathrm{grad}(U)\cdot d\gamma = U(4,1)-U(3,-4) = \frac1{18}-\frac1{26}$$

by the fundamental theorem of calculus / Stokes' theorem, whatever you are using.

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