|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
derivation of directional derivative
|
(Derivation)
|
|
|
Let $f(x,y)$ be a function where $x=x(t)$ and $y=y(t)$ Let $\vec{r}=a\hat{\mathbf{i}}+b\hat{\mathbf{j}}$ be the vector in the desired direction. The line through this vector is given parametrically by:
$\displaystyle x=x_0+at;\quad y=y_0+bt$ endcenter
The derivative of $f$ with respect to $t$ is as follows:
$\displaystyle \frac{\partial f}{\partial t}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$ endcenter
But from the equation of the line, we know that $\frac{dx}{dt}=a$ and $\frac{dy}{dt}=b$ so the derivative becomes:
$\displaystyle \frac{\partial f}{\partial t}=\frac{\partial f}{\partial x}a+\frac{\partial f}{\partial y}b=\nabla f\cdot\vec{r}$ endcenter
|
"derivation of directional derivative" is owned by apmc.
|
|
(view preamble | get metadata)
Cross-references: equation, derivative, line, vector, function
This is version 4 of derivation of directional derivative, born on 2005-07-25, modified 2005-07-27.
Object id is 7267, canonical name is DerivationOfDirectionalDerivative.
Accessed 1614 times total.
Classification:
| AMS MSC: | 26B10 (Real functions :: Functions of several variables :: Implicit function theorems, Jacobians, transformations with several variables) | | | 26B12 (Real functions :: Functions of several variables :: Calculus of vector functions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
