Course: Vector Calculus (MTH 255) (editing)

Multivariable Calculus
Derivatives can be found while holding one or more variables constant

Derivatives of multivariable functions can be found by holding one or more variables constant (subject to physical limitations).
There is a partial derivative in every direction at any point

For a smooth, two dimensional function, a partial derivative can be found at any point and in any direction.
The value of a partial derivative depends on the value(s) of what is held constant

FIXME: Is this really a statement about paths? I don't think we have path-related concepts.
The magnitude of the gradient is proportional to the density of contour lines
Contour Maps

As adjacent contour lines represent equal changes in the function, the denser the contour lines, the quicker the function is changing. Therefore, the magnitude of the gradient is greater where the contour lines are denser.
The components of the gradient are partial derivatives

FIXME
Static Fields
The gradient is a local quantity

The magnitude and direction of the gradient depends on the behavior of the function infinitesimally near the point of interest. One implication of this is that the gradient does not point in the direction of maxima a finite distance away (except sometimes accidentally), but rather in the direction of steepest slope at the point of interest.
lower anchor MTH 255
$d \vec r$ is a small displacement vector Formulas for the Divergence Formulas for the Curl $\int\vec\nabla\cdot\vec F dV= \oint\vec F\cdot d\vec A$ $\oint \vec F\cdot d\vec r = \int \vec\nabla\times\vec F\cdot d\vec A$
The Hill
$\vec \nabla f$
The magnitude of the gradient is proportional to the density of contour lines
The direction of the gradient is perpendicular to contour lines
The gradient is a local quantity The gradient lives in the domain and has the same number of spatial dimensions as the original function
The Valley
The gradient can tell you a small change in a function in any direction (differentials edition)
Divergence
Curl
Stokes' Theorem