Partial derivative of a parametric surface, part 2

TL;DR

Partial derivatives of parametric surfaces help determine tangent vectors and the rate at which changes in input variables cause movement in the output space.

Transcript

• [Voiceover] Hello hello again. So, in the last video, I started talking about how you interpret the partial derivative of a parametric surface function, of a function that has a two variable input, and a three variable vector-valued output. And we typically visualize those as a surface in three dimensional space, and the whole process, I was sayi... Read More

Key Insights

• 🫥 Partial derivatives of parametric surfaces involve visualizing lines representing movement in specific directions and their corresponding changes in the output space.
• ☠️ Tangent vectors derived from partial derivatives indicate the rate of change in the output space based on nudges in the input variables.
• 👾 Visualizing the movement along the curve can help determine the direction of the tangent vector in the output space.
• 🫡 Partial derivatives with respect to different variables can result in different notions of tangent vectors on the surface.
• 💨 Directional derivatives combine partial derivatives in various ways to represent different tangent vectors on the surface.
• 🈸 Partial derivatives of vector-valued functions have applications beyond parametric surfaces.
• 😑 Expressing tangent planes can be done by defining them in terms of two different vectors.

Questions & Answers

Q: How does the animation cheat in representing the t-s plane in the video?

The animation represents the t-s plane on the x-y plane instead of a separate space, making it easier to visualize, although less accurate.

Q: How does the partial derivative with respect to s differ from the partial derivative with respect to t?

The partial derivative with respect to s represents movement in the s direction, while the partial derivative with respect to t represents movement in the t direction.

Q: How are partial derivatives computed for specific cases?

By applying the derivative rules, the partial derivatives can be computed by differentiating each term with respect to the corresponding variable.

Q: How can the direction of movement be determined in the output space?

By observing the increasing direction as the input variable ranges, the corresponding movement along the curve can help determine the direction in the output space.

Summary & Key Takeaways

• Partial derivatives of parametric surfaces involve visualizing how lines representing movement in specific directions correspond to changes in the output space.

• The partial derivative with respect to t represents movement in the t direction, while the partial derivative with respect to s represents movement in the s direction.

• The partial derivative vector is a tangent vector that indicates the rate of change in the output space based on nudges in the input variables.