Multivariable Calculus (Calc III) - Complete Semester Course - Parametric surfaces r(u,v), Multivariable Calculus

A surface is 2-dimensional, so we need two parameters (typically u and v) to parametrize it. Several examples of coming up with parametrizations r(u,v), plus the domains for u and v. Examples include polar coordinates. Multivariable Calculus Unit 3 Lecture 4. The goal is to create descriptions suitable for calculus applications. Important note: - Curves: Being one-dimensional, curves require only one parameter, typically denoted as 𝑡. - Surfaces: As two-dimensional objects, surfaces require two parameters for a complete description. We usually use 𝑢 and 𝑣 as these parameters. Examples of Surface Parametrization 1. Saddle Surface 𝑧=𝑥𝑦: - Parametrization: 𝑟⃗ (𝑢,𝑣)=(𝑢,𝑣,𝑢𝑣), where 𝑢 and 𝑣 are real numbers. - This parametrization captures the essence of the saddle surface, where 𝑧 is the product of 𝑥 and 𝑦. Other examples include: 2. Upper Hemisphere (two ways) 3. Planes 4. Circular Cylinder 𝑥^2+𝑧^2=1 as 𝑟⃗ (𝑢,𝑣)=(cos(𝑢),𝑣,sin(𝑢)), with 𝑢 ranging from 0 to 2𝜋 and 𝑣 being any real number. 5. Paraboloid 𝑧=𝑥^2+𝑦^2 over a unit disc #calculus #multivariablecalculus #mathematics #parametricsurfaces #iitjammathematics #calculus3