Geometry

In this article, we will examine the nature and utility of the first fundamental form, a quadratic form on a surface. Given a curve parametrized by \(u=u(t)\) and \(v=v(t)\) which lies on a surface \(X\) parametrized by \(X=X(u,v)\) we have that $$ ds = |\frac{dX}{dt}| dt = |X_u \frac{du}{dt} + X_v \frac{dv}{dt}| dt = \sqrt{(X_u \dot{u} + X_v \dot{v}) \cdot (X_u \dot{u} + X_v \dot{v})} $$ Now defining the coefficients, called the First Fundamental Form coefficients E, F and G, we have

For a set of orthonormal vectors \((v_1,v_2,...,v_m)\) in an m-dimensional vector space , and set of constants \(r_1,r_2,...r_m \in \mathbb{R}\) , if we consider the equation \(r_1v_1+r_2v_2+...+r_mv_m=0\) with the objective of demonstrating that the vectors in \((v_1,v_2,...,v_m)\) are linearly independent, then we must show that \(r_1=r_2=...=r_m=0\) . If we take the inner product on both sides of the equation , we have that $$ <r_1v_1+r_2v_2+…+r_mv_m, v_1> = <0,v_1> $$ Using the properties of inner products, we have that