Let mathbf F=langle 2xy, x^2rangle=∇(x^2y). Compute ∫_Cmathbf F· dmathbf r for any curve C from (0,0) to (1,2).

Exercise · ZW-CONSERVATIVE-FIELD-003 · ★☆☆ Basic

Let \(\mathbf F=\langle 2xy,\,x^2\rangle=\nabla(x^2y)\). Compute \(\int_C\mathbf F\cdot d\mathbf r\) for any curve \(C\) from \((0,0)\) to \((1,2)\).

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By the fundamental theorem, the integral is \(f(1,2)-f(0,0)=1^2\cdot2-0=2\), whatever the curve.

A vector field \(\mathbf F\) is conservative if it is the gradient of a scalar function \(f\), called a potential function of \(\mathbf F\).

\[\mathbf F=\nabla f\]