# Difference between revisions of "Gradient"

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(새 문서: {{키워드 |관련코스=다변수미적분학 |소분류=수학 }} ===Definition=== :::<math> \nabla f = \cfrac{1}{h_i}{\partial f \over \partial q^i} \mathbf{e}^i </math> :If we...) |
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{{키워드 | {{키워드 | ||

− | |관련코스= | + | |관련코스=다변수 미적분학 |

− | |소분류=수학 | + | |소분류=수학, 물리 |

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===Definition=== | ===Definition=== |

## Latest revision as of 21:00, 1 September 2020

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]