Gradient

Assume \displaystyle{(x, y)} represents the position of an object and \displaystyle{f(x,y)} is a scalar field on the \displaystyle{x}\displaystyle{y} plane. Then \displaystyle{\frac{\partial f}{\partial x}} represents the change of \displaystyle{f} per unit length along the positive \displaystyle{x} direction. In other words, it is the spatial rate of change of \displaystyle{f} along the \displaystyle{x} direction.

Similarly, derivative \displaystyle{\frac{\partial f}{\partial y}} represents the spatial rate of change of \displaystyle{f} along the \displaystyle{y} direction.

For an arbitrary direction, due to the nature of displacement, the change of \displaystyle{f} is \displaystyle{\delta f = \frac{\partial f}{\partial x} \delta x + \frac{\partial f}{\partial x} \delta y} when the object has finished moving \displaystyle{\delta x} in \displaystyle{x} direction and then \displaystyle{\delta y} in \displaystyle{y} direction.

Then, the spatial rate of change of \displaystyle{f} is

\displaystyle{   \begin{aligned}   &\frac{\delta f}{\sqrt{(\delta x)^2 + (\delta y)^2}} \\  &= \frac{\partial f}{\partial x} \frac{\delta x}{\sqrt{(\delta x)^2 + (\delta y)^2}}  + \frac{\partial f}{\partial x} \frac{\delta y}{\sqrt{(\delta x)^2 + (\delta y)^2}} \\  \end{aligned} }

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For simplicity, denote the resultant displacement as \displaystyle{\mathbf{v}}:

\displaystyle{\mathbf{v} = (\delta x, \delta y)}

and define \displaystyle{\nabla f(x)} as

\displaystyle{\left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)}

Then, the change of the \displaystyle{f} due to the displacement \displaystyle{\mathbf{v}} is

\displaystyle{\begin{aligned}  \left(\delta f\right)_{\mathbf{v}}  &= \frac{\partial f}{\partial x} \delta x + \frac{\partial f}{\partial x} \delta y \\  &= \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial x}\right) \cdot (\delta x, \delta y) \\  &= \left(\nabla f\right) \cdot \mathbf{v} \\  \end{aligned}}

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So the spatial rate of change \displaystyle{f} along the direction of the vector \displaystyle{\mathbf{v}} is

\displaystyle{\begin{aligned}  D_{\mathbf{v}}(f)  &= \frac{\left(\delta f\right)_{\mathbf{v}}}{|\mathbf{v}|} \\  &= \frac{\partial f}{\partial x} \frac{\delta x}{\sqrt{(\delta x)^2 + (\delta y)^2}}  + \frac{\partial f}{\partial x} \frac{\delta y}{\sqrt{(\delta x)^2 + (\delta y)^2}} \\  &= \left(\nabla f\right) \cdot \frac{\mathbf{v}}{|\mathbf{v}|} \\  &= \left(\nabla f\right) \cdot \hat{\mathbf{v}} \\  \end{aligned}}

\displaystyle{D_{\mathbf{v}}(f)} is called directional derivative.

— Me@2016-02-06 09:49:22 PM

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This is the reason that \displaystyle{\nabla f} is in the steepest direction.

If \displaystyle{\hat{\mathbf{v}}} is chosen to be parallel to \displaystyle{\nabla f}, the directional derivative \displaystyle{\left(\nabla f\right) \cdot \hat{\mathbf{v}}} would be maximized.

— Me@2021-08-20 05:20:02 PM

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2016.02.21 Sunday (c) All rights reserved by ACHK