Example · Undergraduate
Terms
| Term | This example | Why |
|---|
partial_derivative |
$\frac{\partial f}{\partial x} = 2x$ and $\frac{\partial f}{\partial y} = 3$
|
|
| gradient |
$\nabla f(x, y) = (2x, 3)$
|
A vector.
|
| Jacobian |
$Df(x, y) = \begin{bmatrix} 2x & 3 \end{bmatrix}$
|
A $1 \times 2$ matrix, the transpose of the gradient.
|
| directional derivative |
$D_v f(x, y) = \nabla f(x, y) \cdot v = 2x$, in the direction $v = (1, 0)$
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$\nabla f \cdot (1, 0) = 2x$.
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| differential |
$\mathrm{d}f_{(x, y)} : (v_1, v_2) \mapsto 2x v_1 + 3 v_2$
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A linear map that takes a direction and returns a rate of change.
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| Lie derivative |
$L_F f(x, y) = \nabla f(x, y) \cdot F(x, y) = -2x^2$, along the vector field $F(x, y) = (-x, 0)$
|
$\nabla f \cdot F = 2x \cdot (-x) + 3 \cdot 0$. The direction varies from point to point.
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Distinctions using this example
- Types of multivariable derivatives: The gradient of $f$ is a vector, its Jacobian is the $1 \times 2$ matrix with the same entries, and its differential is the linear map given by that matrix. The directional derivative needs a direction $v$, and the Lie derivative uses a direction $F(x, y)$ that varies from point to point.
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