Gradient

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Gradient
관련코스 다변수 미적분학
소분류 수학, 물리
선행 키워드
연관 키워드

Definition

[math] \nabla f = \cfrac{1}{h_i}{\partial f \over \partial q^i} \mathbf{e}^i [/math]
If we have a 3-dimensional function [math]f(x,y,z)[/math], the derivative will have to give us the relationship between [math]df[/math] and [math]d\vec{r} = (dx,dy,dz)[/math].
This kind of derivative is called gradient.
As we know [math]df[/math] can be represented as below using partial derivatives:
[math]df = \left(\frac{\partial f}{\partial x}\right)dx + \left(\frac{\partial f}{\partial y}\right)dy + \left(\frac{\partial f}{\partial z}\right)dz [/math]
So, in Cartesian coordinates, we can define gradient as follows
[math]\nabla f \equiv \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)[/math]
so we can write [math]df = \nabla f \cdot d\vec{r} [/math]

Cylindrical coordinates

[math]\nabla f={\partial f \over \partial \rho}\hat{\boldsymbol \rho} + {1 \over \rho}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi} + {\partial f \over \partial z}\hat{\mathbf z}[/math]

Spherical coordinates

[math]\nabla f={\partial f \over \partial r}\hat{\mathbf r}+ {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol \theta}+ {1 \over r\sin\theta}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi}[/math]