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directional derivative
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Definition/Summary
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The directional derivative of a scalar field [itex]f[/itex], in the direction [itex]\mathbf{u}[/itex] is defined thus,
[tex]\nabla_{u}f= \nabla f\cdot\frac{\mathbf{u}}{\left|\mathbf{u}\right|}[/tex]
That is, the directional derivative is defined as the scalar product of the gradient of [itex]f[/itex] and the unit vector [itex]\mathbf{u}/|\mathbf{u}|[/itex]. |
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Equations
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In Cartesian coordinates,
[tex]\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)[/tex]
where [itex]\mathbf{u} =\left(u_x,u_y,u_z\right)[/itex]. |
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Recent forum threads on directional derivative
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Breakdown
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Mathematics
> Calculus/Analysis
>> Calculus
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Extended explanation
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| The directional derivative of a scalar field [itex]f[/itex], in the direction [itex]\bold{u}[/itex] represents that rate at which [itex]f[/itex] changes in the given direction and is analogous to usual derivative. |
Commentary
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