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Mathematics LibreTexts

3.4: Grad, curl and div

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    6491
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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 = \(M_x + N_y\).

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