Surfaces and Partial Differentiation

In 3D, a surface can be described by all the points such that or for some constant . The plane is horizontal and the -axis is vertical.

A visualisation of :

Sections

A section is a cross-section of the surface for specific values of or : or for constants or .
A visualisation of sections through :

Contours

A contour is a curve where the value of is fixed: for some constant .
A visualisation of contours of , with :

Partial differentiation

For a surface defined as:

The partial derivatives of are denoted as:

: differentiate with respect to , treating as constant.
: differentiate with respect to , treating as constant.

Higher-order partial derivatives are denoted as:

^[1]
^[1-1]

The mixed derivative theorem states that for most well-behaved continuous functions:

Stationary points

A stationary point is where and . These are classified as minima, maxima, or saddle points, depending on the determinant of the Hessian matrix:

If and , there is a minimum.
If and , there is a maximum.
If there is a saddle point.
If the nature of the stationary point cannot be determined by this test.

Tangent planes

The tangent plane to a surface at the point is given by:

Footnote on order of mixed derivatives
This is the order that the textbook insists is correct, whilst the rest of the Internet disagrees. Practically, it doesn't matter as on this course, all functions will likely satisfy the mixed derivative theorem. This may be corrected soon pending further investigation (the author is severely sleep deprived).↩︎↩︎