## Lecture: Multivariable Differentials

Surfaces/Bridge Workshop 2023
This mini-lecture demonstrates the relationship between $df$ on the tangent plane to its “components“ in coordinate directions, leading to the multivariable chain rule.
What students learn
• Visualization of $df$ in the tangent plane;
• Relationship of $df$ to small changes in coordinates, such as $dx$ and $dy$;
• Multivariable chain rule
Use a small whiteboard to represent the tangent plane to the graph of a function of 2 variables. A small change in position in the domain corresponds to a diagonal line in the tangent plane, representing $df$. To get from one end to the other, one can first proceed a distance $dx$ in the $x$ direction, then a distance $dy$ in the $y$ direction, with corresponding changes to $f$. This triangle can be drawn on the whiteboard. These changes can be represented as the slope in the corresponding direction, multiplied by the distance, that is, as $\frac{\partial f}{\partial x}dx$, and similarly for $y$. The change $df$ in $f$ is of course the sum of these two changes in the coordinate directions, leading to the multivariable chain rule in differential form, namely $df = \frac{\partial f}{\partial x}\,dx+\frac{\partial f}{\partial y}\,dy$
Keywords
Relationships between differentials tangent plane and partial derivatives.
Learning Outcomes