Gradient

Assume \((x, y)\) represents the position of an object and \(f(x,y)\) is a scalar field on the \(x\)-\(y\) plane.

Then \(\frac{\partial f}{\partial x}\) represents the change of \(f\) per unit length along the positive \(x\) direction. In other words, it is the spatial rate of change of \(f\) along the \(x\) direction.

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

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

Then, the spatial rate of change of \(f\) is\[\begin{align}  &\frac{\delta f}{\sqrt{(\delta x)^2 + (\delta y)^2}} \\
&= \frac{\partial f}{\partial x} \frac{\delta x}{\sqrt{(\delta x)^2 + (\delta y)^2}}
+ \frac{\partial f}{\partial x} \frac{\delta y}{\sqrt{(\delta x)^2 + (\delta y)^2}} \end{align}\]

For simplicity, denote the resultant displacement as \(\mathbf{v}\):\[\mathbf{v} = (\delta x, \delta y)\]and define \(\nabla f(x)\) as\[\nabla f(x) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)\]
Then, the change of the \(f\) due to the displacement \(\mathbf{v}\) is\[\begin{align} 
\left(\delta f\right)_{\mathbf{v}}
      &= \frac{\partial f}{\partial x} \delta x + \frac{\partial f}{\partial x} \delta y \\ 
      &= \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial x}\right) \cdot (\delta x, \delta y) \\
      &= \left(\nabla f\right) \cdot \mathbf{v} \\
\end{align}\]So the spatial rate of change \(f\) along the direction of the vector \(\mathbf{v}\) is
\[\begin{align}
D_{\mathbf{v}}(f)
&= \frac{\left(\delta f\right)_{\mathbf{v}}}{|\mathbf{v}|} \\
&= \frac{\partial f}{\partial x} \frac{\delta x}{\sqrt{(\delta x)^2 + (\delta y)^2}} 
 + \frac{\partial f}{\partial x} \frac{\delta y}{\sqrt{(\delta x)^2 + (\delta y)^2}} \\
&= \left(\nabla f\right) \cdot \frac{\mathbf{v}}{|\mathbf{v}|} \\
&= \left(\nabla f\right) \cdot \hat{\mathbf{v}} \\
\end{align}\]
\(D_{\mathbf{v}}(f)\) is called directional derivative.

“In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the coordinate curves, all other coordinates being constant.”

— Wikipedia on Directional derivative

— Me@2016-02-06 09:49:22 PM

2016.02.21 Sunday (c) All rights reserved by ACHK