Tensor Calculus Lecture 10b: The Normal Derivative

This paragraph introduces the concept of normal derivatives, which the lecturer realized was omitted from previous lectures. It is highlighted as an essential topic with significant applications in integration and the divergence theorem. The lecturer sets the stage for the derivation of Gauss's theorem in the next lecture, emphasizing the importance of understanding normal derivatives in both a physical and geometric context. The paragraph concludes with a preliminary definition of the normal derivative in the context of a three-dimensional Euclidean space.

The lecturer delves into a more precise definition of the normal derivative, explaining how it is measured along a straight line normal to the surface at a given point. The importance of the Euclidean nature of the ambient space is discussed, and the lecturer invites the audience to consider how the definition might change in a non-Euclidean space. The paragraph concludes with a parameterization of the straight line and an exploration of how the normal derivative relates to the function defined on the surface and in the ambient space.

The lecturer chooses Cartesian coordinates for simplicity and expedience, despite acknowledging it as a 'wrong' approach in the spirit of tensor calculus. A detailed parameterization of the straight line in Cartesian coordinates is provided, and the connection between the function defined on the surface and in the ambient space is established. The paragraph culminates in the derivation of the normal derivative as the dot product of the unit normal vector and the gradient of the function.

Building on the previous discussion, the lecturer explores the second normal derivative, cautioning against common mistakes in differentiation. The process involves evaluating the second derivative of the function with respect to the parameter s, and the lecturer emphasizes the importance of doing so at the correct point. The paragraph concludes with an expression for the second normal derivative that includes the second-order covariant derivative and the properties of the ambient space.

The lecturer contemplates the implications of defining normal derivatives on non-Euclidean spaces, suggesting that geodesics might replace straight lines in such contexts. There is a brief discussion on how the definition of the second normal derivative might be affected, hinting at the need for a more general approach that does not rely on the specific properties of Cartesian or Euclidean spaces.

The relationship between the ambient Laplacian of a function defined in the entire space and its surface Laplacian is examined. The lecturer uses a comprehensive approach that involves the chain rule, the derivative of the metric tensor, the mean curvature, and the projection formula. The paragraph showcases the application of various elements of differential geometry, culminating in an expression that relates the surface Laplacian to the ambient Laplacian.

The lecturer reflects on the derivation process, acknowledging the expedience of using Cartesian coordinates but recognizing the need for a more general and coordinate-invariant approach. The paragraph discusses the importance of tensor calculus and its coordinate invariance, hinting at a future correction of the approach to align with the principles of tensor calculus.

In the final paragraph, the lecturer summarizes the key takeaways from the lecture, emphasizing the importance of the calculations performed and the elements of differential geometry involved. The paragraph concludes with a note of personal urgency, as the lecturer must leave, but not before expressing gratitude to the audience for their attention and promising to continue the discussion in a future lecture.