Multivariable Calculus (Calc III) - Complete Semester Course - The Chain Rule in Multivariable Calculus

The Chain Rule in two particular situations: (1) evaluating the composition of a scalar-valued function along a curve, and (2) evaluating f(x,y) with x and y functions of s and t. (Multivariable Calculus Unit 3 Lecture 13) We first consider a vector-valued function 𝑟⃗ of a single variable 𝑡, and a scalar-valued function 𝑓 of 𝑛 inputs: 𝑓∘𝑟⃗ : 𝑡 ↦ 𝑟⃗ (𝑡) ↦ 𝑓∘𝑟⃗ (𝑡) = 𝑓(𝑟⃗ (𝑡)). We see that overall the composition 𝑓(𝑟⃗ (𝑡)) is a function mapping real numbers to real numbers. Its derivative is obtained using the chain rule, which in this context translates to the gradient of 𝑓 evaluated at 𝑟⃗ (𝑡), dotted with the derivative 𝑟⃗ ′(𝑡), the velocity vector: 𝑑/𝑑𝑡(𝑓∘𝑟⃗ (𝑡))=∇𝑓(𝑟⃗ (𝑡))⋅𝑟⃗ ′(𝑡). For example, let 𝑟⃗ (𝑡)=⟨𝑡cos(𝑡),𝑡sin(𝑡)⟩ and 𝑓(𝑥,𝑦)=sqrt(𝑥^2+𝑦^2). We find the derivative of 𝑓(𝑟⃗ (𝑡)) with respect to 𝑡 using two methods: direct computation of the composition followed by differentiation, and application of the chain rule. Both methods yield consistent results, validating the chain rule's application. In a second example, we let 𝑟⃗ (𝑡)=⟨4cos(𝑡),3sin(𝑡),𝑡^2⟩ and 𝑓(𝑥,𝑦,𝑧)=𝑥^2+𝑦^2+𝑧^2. By applying the chain rule, we calculate the derivative of 𝑓(𝑟⃗ (𝑡)) with respect to 𝑡. This involves finding the gradient of 𝑓, evaluating it along 𝑟⃗ (𝑡), and then dotting it with the velocity vector 𝑟⃗ ′(𝑡). We explore a situation where a scalar-valued function 𝑓(𝑥,𝑦) depends on 𝑥 and 𝑦, which are functions of other variables 𝑠 and 𝑡. The chain rule enables us to compute the partial derivatives of 𝑓 with respect to 𝑠 and 𝑡, considering how 𝑥 and 𝑦 change with respect to these variables: ∂𝑓∂𝑠=∂𝑓/∂𝑥 ∂𝑥/∂𝑠 + ∂𝑓/∂𝑦 ∂𝑦∂/𝑠 ∂𝑓/∂𝑡=∂𝑓/∂𝑥 ∂𝑥/∂𝑡 + ∂𝑓/∂𝑦 ∂𝑦/∂𝑡. #mathematics #multivariablecalculus #calculus #differentiation #chainrule #partialderivatives #iitjammathematics #calculus3