In 3D, a surface can be described by all the points such that
A visualisation of
A section is a cross-section of the surface for specific values of
A visualisation of sections through
A contour is a curve where the value of
A visualisation of contours of
For a surface defined as:
The partial derivatives of
Higher-order partial derivatives are denoted as:
The mixed derivative theorem states that for most well-behaved continuous functions:
When
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).↩︎↩︎