Partials

We find partials when we are looking at equations used in a discussion of contravariance and covariance.

\begin{bmatrix} Contravariance && Covariance \\ \tilde A^i = \dfrac {\partial \tilde x^i} {\partial x^j} A^j && \tilde A_i = \dfrac {\partial x^j} {\partial \tilde x^i} A^j \end{bmatrix}

Regarding partials, a superscript in the denominator is equivalent to a subscript in the numerator.

We can perform partial differentiation on a multi-variable function and get a slice through it depending on our choices for the two variables shown in the partial.

This gets interesting, your multivariable function to show curvature, yet when you slice in the direction of a coordinate your slice is linear.

Aftet two drinks, you could do partial differentiation on a single variable function making your slice follow the x-coordinate. To explain it informally, you are slicing a slice.

Appendic C

Should we go back to Calculus 3 and find all the bases that you learned about partial differentiation?