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# 3.4: Grad, curl and div

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Gradient. For a function $$f(x, y)$$, the gradient is defined as grad$$f = \nabla f = (f_x, f_y)$$. A vector field F which is the gradient of some function is called a gradient vector field.

Curl. For a vector in the plane F$$(x, y) = (M(x, y), N(x, y))$$ we define

curlF = $$N_x - M_y$$.

Note. The curl is a scalar. In general, the curl of a vector field is another vector field. However, for vectors fields in the plane the curl is always in the $$\widehat{k}$$ direction, so we have simply dropped the $$\widehat{k}$$ and made curl a scalar.

Divergence. The divergence of the vector field F $$= (M, N)$$ is

divF = $$M_x + N_y$$.