# 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.

### :clapper: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.

{% embed url="<https://youtu.be/9r_66nKtMnA>" %}

### :pen\_ballpoint:Mathematical Proof

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