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.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

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Parametric vs. Implicit Forms: A 3D surface can be represented in multiple ways. The solutions show how to work with surfaces defined parametrically, using a position vector $r(t, s)$, and implicitly, using a scalar function $\phi(x, y, z)=C$.
Tangent Vectors Define a Surface: The partial derivatives of a position vector, $\frac{\partial r}{\partial t}$ and $\frac{\partial r}{\partial s}$, produce vectors that are tangent to the surface.
Two Ways to Find a Normal Vector: A vector normal (perpendicular) to a surface can be found in two ways:
1. Using a Cross Product: By taking the cross product of the two tangent vectors from the parametric representation, i.e., $N=\frac{\partial r}{\partial t} \times \frac{\partial r}{\partial s}$.
2. Using the Gradient: By calculating the gradient of the implicit function, $\nabla \phi$.
Parallelism is a Key Property: A fundamental result of vector calculus is that the normal vector found from the cross product and the gradient vector are always parallel. This parallelism is successfully verified for all three surfaces: the plane, the paraboloid, and the corrugated sheet.

🎬Demonstration

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.

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