Multivariable Calculus (Calc III) - Complete Semester Course - Introduction to polar coordinates and integration, Multivariable Calculus

We review polar coordinates in the (x,y)-plane. Then we discuss the piece we need to set up a double Riemann sum and look at the derivation of the area term r dr dθ. (Unit 5 Lecture 1) We explore the advantages of switching from 𝑥, 𝑦 coordinates to polar coordinates for certain types of problems. This transition is particularly beneficial when dealing with circular or annular regions. We proceed to review the fundamentals of polar coordinates. The coordinates (𝑥,𝑦) are connected to polar coordinates through the equations 𝑥=𝑟cos(𝜃) and 𝑦=𝑟sin(𝜃), where 𝑟 is the displacement from the origin, and 𝜃 is the angle from the positive x-axis. We emphasize the importance of the Pythagorean identity 𝑥^2+𝑦^2=𝑟^2. We note that polar coordinates are not unique. For example, the angle 𝜃 can be expressed in multiple ways, such as 𝜃=𝜋/3 or 𝜃=𝜋/3+2𝜋. Traditionally, we consider 𝜃 to be between 0 and 2𝜋, with 𝑟 representing a positive distance from the origin. [To base your conversion around the point (ℎ,𝑘) rather than the origin, the conversion is 𝑥=ℎ+𝑟cos(𝜃), 𝑦=𝑘+𝑟sin(𝜃). In this case, the identity is (𝑥−ℎ)^2+(𝑦−𝑘)^2=𝑟^2.] We then derive the form for a double integral in polar coordinates, focusing on the differentials needed for integration. We compare the differentials 𝑑𝑥 𝑑𝑦 in rectangular coordinates with those in polar coordinates, aiming to establish a new form. We situation this computation in the context of integrating a function 𝑤=𝑓(𝑥,𝑦) over a domain that is more naturally described in polar coordinates, specifically an annulus. Our strategy involves chopping the domain into small pieces, not parallel to the 𝑥 and 𝑦 axes, but rather in a manner resembling slicing a pie. We identify small changes in angles (Δ𝜃) and radii (Δ𝑟), leading to the partition of the domain into small sections for the integration. We compute the area of a pie-shaped wedge in polar coordinates using the philosophy of "good = all - bad." We calculate the areas of larger and smaller circular sectors and subtract them to find the area of interest. This leads us to a formula involving the sum of radii and the change in radius. Ultimately we find that areas scale up as our radius increases, leading to a conversation factor of 𝑟: 𝑟 Δ𝑟 Δ𝜃 → 𝑟 𝑑𝑟 𝑑𝜃. We do one example of computing the mass of a lamina shaped like an annulus in the 𝑥𝑦-plane. The lamina is annular, with inner radius 1 and outer radius 3, and the density at any point (𝑥,𝑦) is given by the density function 𝜎(𝑥,𝑦)=𝑥^2+𝑦^2. #mathematics #math #multivariablecalculus #doubleintegrals #doubleintegration #polarcoordinates #iitjammathematics #calculus3 #IntegrationTechniques #MathEducation #cylindrical