At each point on the curve, we can define a plane (called the osculating plane) in which the curve locally lies (e.g., its tangent vector lies in this plane). The quantity κ is a measure of how the tangent vector is changing in the plane, and τ is a measure of how much the curve is twisting out of the osculating plane.

One sees that the tangent space or tangent plane to S at p, which is defined to consist of all tangent vectors to S at p, is a two-dimensional linear subspace of ℝ 3; it is often denoted by T p S. The normal space to S at p , which is defined to consist of all normal vectors to S at p , is a one-dimensional linear subspace of ℝ 3 which is ...

View space_curves.pdf from MATH 53 at University of California, Berkeley. Math 53 - 2024 1 TANGENT VECTORS Space curves and Tangent vectors 1 Tangent Vectors 1. (Some scaffolding) We wrote the

The second-order Cauchy stress tensor describes the stress experienced by a material at a given point. For any unit vector , the product is a vector, denoted (), that quantifies the force per area along the plane perpendicular to .This image shows, for cube faces perpendicular to ,,, the corresponding stress vectors (), (), along those faces. Because the stress tensor takes one vector as input ...

In Sect. 1.3 it was seen that, in the case the fluid is in a non-inertial reference frame, Newton's equations of motion must include, in the balance of forces, appropriate apparent forces. In the case of geophysical fluids, the apparent force to consider for motions with sufficiently large time scale, so that the rotation affects the dynamics, is the Coriolis force.