🧄Surface Parametrisation and the Verification of the Gradient-Normal Relationship

A surface can be represented either parametrically using a position vector $r(t, s)$ or implicitly with a function $\phi(x, y, z)=C$. The partial derivatives of the parametric form yield tangent vectors, and their cross product gives a normal vector. Crucially, this normal vector is always parallel to the gradient vector of the implicit function, a key principle of vector calculus verified across different surface types.

🎬the relationship between tangent vectors and the normal vector and the gradient vector of a 3D surface

This 3D interactive demo visualizes key concepts in vector calculus by showing the relationship between a surface's tangent, normal, and gradient vectors. You can select from three different surfaces—a plane, a paraboloid, or a corrugated sheet—and see how the tangent vectors define the surface's local orientation. The app then calculates the normal vector via a cross product and the gradient vector from the surface's implicit function, confirming they are parallel. Real-time display of vector values and their dot product reinforces the theoretical relationship, providing a powerful educational tool for understanding these fundamental principles.

🖊️Mathematical Proof

Surface Parametrisation and the Verification of the Gradient-Normal Relationship | Proof and Derivationviadean on Notion