Multivariable Calculus (Calc III) - Complete Semester Course - Introduction to line integrals, Multivariable Calculus

We mention the two types of line integrals ("scalar" and "vector"). These integrals vary in what is being integrated (a scalar-valued function or a vector-valued function) and their geometric and contextual applications. The name "line integral" is misleading, as our computations are usually integration over curves rather than just straight lines. Both need to start with a parametrization r(t) for the curve we integrate over. We discuss parametrizing both plane and space curves using vector-valued functions 𝐫(𝑡). For example, to parametrize a section of the unit circle from (1,0) to (0,1), we use 𝐫(𝑡)=⟨cos(𝑡), sin(𝑡)⟩ for 0 ≤ 𝑡 ≤ 𝜋/2. Another example is parametrizing a parabola 𝑦=𝑥^2 from (0,0) to (3,9), using 𝐫(𝑡)=⟨𝑡, 𝑡^2⟩ for 0 ≤ 𝑡 ≤ 3. For more examples of curve parametrizations, please see https://youtu.be/neVWAipZQzw. Recall that for a curve parametrized by 𝐫(𝑡), the arc length function from 𝑡=𝑎 to a variable upper limit 𝑡 is given by 𝑠(𝑡)=∫_𝑎^t ‖𝐫′(𝑢)‖ 𝑑𝑢. This function allows us to relate the differential arclength 𝑑𝑠 to the differential parameter 𝑑𝑡, crucial for setting up scalar line integrals. From the Fundamental Theorem of Calculus, 𝑑𝑠.𝑑𝑡=‖𝐫′(𝑡)‖. Therefore, the relationship between the differentials 𝑑𝑠 and 𝑑𝑡 (where 𝑑𝑡 refers to a specific parametrization) is 𝑑𝑠=‖𝐫′(𝑡)‖𝑑𝑡. (This equation is analogous to relating 𝑑𝑥 and 𝑑𝑢 when 𝑢-sub.) Now to put everything together, in scalar line integrals, we integrate a scalar-valued function 𝑓 over a curve. The curve is represented by a parametrization 𝐫(𝑡). The pullback of a function 𝑓 by the parametrization 𝐫(𝑡) is an essential concept in setting up the scalar line integral. This pullback is defined as the composition 𝑓(𝐫(𝑡)). Here's what happens in this process: Composition of Functions: The composition 𝑓(𝐫(𝑡)) means we first apply the parametrization 𝐫(𝑡) and then apply the function 𝑓 to the result. This process evaluates 𝑓 along the curve defined by 𝐫(𝑡). Mathematical Formulation: Suppose 𝐫(𝑡)=⟨𝑥(𝑡),𝑦(𝑡)⟩ for a curve in the plane. Then the pullback 𝑓(𝐫(𝑡)) is given by 𝑓(𝐫(𝑡))=𝑓(𝑥(𝑡),𝑦(𝑡)). The scalar line integral of 𝑓 along the curve defined by 𝐫(𝑡) from 𝑡=𝑎 to 𝑡=𝑏 is: ∫_𝐶 𝑓 𝑑𝑠 = ∫_𝑏^𝑎𝑓 (𝐫(𝑡)) ⋅ ‖𝐫′(𝑡)‖ 𝑑𝑡. Here, ‖𝐫′(𝑡)‖ represents the speed of the curve at time 𝑡, and the integral computes the accumulated value of 𝑓 along the curve. Notice 𝑑𝑠 ↦ ‖𝑟⃗ ′(𝑡)‖𝑑𝑡. This process transforms integrating a function over a curve into integrating a composition of functions over a simple interval, like in Calculus I. (This is Multivariable Calculus, Unit 6 Lecture 4, it is a warm-up for our next lecture on line integrals.) #calculus #multivariablecalculus #mathematics #lineintegral #iitjammathematics #calculus3 #lineintegrals #vectorcalculus #mathtutorial