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directional derivative

 Definition/Summary The directional derivative of a scalar field $f$, in the direction $\mathbf{u}$ is defined thus, $$\nabla_{u}f= \nabla f\cdot\frac{\mathbf{u}}{\left|\mathbf{u}\right|}$$ That is, the directional derivative is defined as the scalar product of the gradient of $f$ and the unit vector $\mathbf{u}/|\mathbf{u}|$.

 Equations In Cartesian coordinates, $$\nabla_u f\left(x,y,z\right) = \frac{1}{\left|\mathbf{u}\right|}\left(\frac{\partial f}{\partial x}u_x + \frac{\partial f}{\partial y}u_y + \frac{\partial f}{\partial z}u_z\right)$$ where $\mathbf{u} =\left(u_x,u_y,u_z\right)$.

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 Breakdown Mathematics > Calculus/Analysis >> Calculus

 Extended explanation The directional derivative of a scalar field $f$, in the direction $\bold{u}$ represents that rate at which $f$ changes in the given direction and is analogous to usual derivative.