Multivariable Calculus (Calc III) - Complete Semester Course - Scalar surface integrals, Multivariable Calculus

We look at integrating w=f(x,y,z) over surfaces, including computing the mass of a lamina. We begin by considering the process of integrating over a domain which is part of a surface. The differential for this process is 𝑑𝑆, representing a small piece of surface area. We relate this to the differentials 𝑑𝑢 and 𝑑𝑣, with 𝑑𝑆 being the magnitude of the cross product of 𝑟⃗_𝑢 and 𝑟⃗_𝑣, 𝑑𝑆=‖𝑟⃗_𝑢 × 𝑟⃗_𝑣‖ 𝑑𝑢 𝑑𝑣. (Unit 6 Lecture 13) For a scalar-valued function 𝑓(𝑥,𝑦,𝑧), continuous and defined on a surface 𝑆 with a parametrization 𝑟⃗(𝑢,𝑣), the surface integral of 𝑓 over 𝑆 is written as the double integral of 𝑓 with respect to 𝑑𝑆. Computationally, we integrate over a domain 𝐷 of 𝑓(𝑟⃗(𝑢,𝑣)) times ‖𝑟⃗_𝑢×𝑟⃗_𝑣‖𝑑𝑢𝑑𝑣. To summarize, ∬_𝑆 𝑓 𝑑𝑆 = ∬_𝐷 𝑓(𝑟⃗ (𝑢,𝑣)) ‖𝑟⃗ 𝑢×𝑟⃗ 𝑣‖ 𝑑𝑢 𝑑𝑣. Example 1 Setup: Paraboloid Surface Integral We set up the surface integral of 𝑓(𝑥,𝑦,𝑧)=𝑥𝑦 over the portion of the paraboloid 𝑧=9−𝑥2−𝑦2 that lies above the plane 𝑧=5 and in the first octant. We can use polar coordinates to parametrize this surface: 𝑟⃗ (𝑢,𝑣) = ⟨𝑢cos𝑣, 𝑢sin𝑣, 9−𝑢^2⟩. Example 2 Setup: Mass of a Lamina To compute the mass of a lamina occupying a region in ℝ3 described by 𝑧=sqrt(16−𝑥^2−𝑦^2) and 𝑧≥2 with density function 𝜎(𝑥,𝑦,𝑧)=𝑧, we parameterize the surface using spherical coordinates. With our parameters 𝜃 and 𝜙, we set bounds from 0 to 2𝜋 and from 0 to 𝜋3, respectively. The parametrization is 𝑟⃗ (𝑢,𝑣) = ⟨4cos𝑢sin𝑣, 4sin𝑢sin𝑣, 4cos𝑣⟩. #mathematics #mathematics #multivariablecalculus #surfaceintegral #doubleintegrals #iitjammathematics #calculus3 #vectorcalculus #mathtutorial