Question

Find \(\mathbf{F}^{\prime}\) and \(J \mathbf{F}\). (a) \(\mathbf{F}(r, \theta)=\left[\begin{array}{l}r \cos \theta \\ r \sin \theta\end{array}\right]\) (b) \(\mathbf{F}(r, \theta, \phi)=\left[\begin{array}{c}r \cos \theta \cos \phi \\\ r \sin \theta \cos \phi \\ r \sin \phi\end{array}\right]\) (c) \(\mathbf{F}(r, \theta, z)=\left[\begin{array}{c}r \cos \theta \\ r \sin \theta \\ z\end{array}\right]\)

Short answer

Question: Compute the first derivatives and the Jacobian matrices for the following function vectors: a) F(r, θ) = [r cosθ, r sinθ] b) F(r, θ, φ) = [r cosθ cosφ, r sinθ cosφ, r sinφ] c) F(r, θ, z) = [r cosθ, r sinθ, z] Answer: a) F'(r, θ) = [cosθ, -r sinθ, sinθ, r cosθ]; JF(r, θ) = [[cosθ, -r sinθ], [sinθ, r cosθ]] b) F'(r, θ, φ) = [cosθ cosφ, -r sinθ cosφ, -r cosθ sinφ, sinθ cosφ, r cosθ cosφ, -r sinθ sinφ, sinφ, 0, r cosφ]; JF(r, θ, φ) = [[cosθ cosφ, -r sinθ cosφ, -r cosθ sinφ], [sinθ cosφ, r cosθ cosφ, -r sinθ sinφ], [sinφ, 0, r cosφ]] c) F'(r, θ, z) = [cosθ, -r sinθ, 0, sinθ, r cosθ, 0, 0, 0, 1]; JF(r, θ, z) = [[cosθ, -r sinθ, 0], [sinθ, r cosθ, 0], [0, 0, 1]]

Step-by-step solution

(a) Find F'(r, θ) and JF(r, θ) for F(r, θ) = [r cosθ, r sinθ]

First, find the partial derivatives of F(r, θ) with respect to r and θ: F'(r, θ) = [∂(r cosθ)/∂r, ∂(r cosθ)/∂θ, ∂(r sinθ)/∂r, ∂(r sinθ)/∂θ] F'(r, θ) = [cosθ, -r sinθ, sinθ, r cosθ] Next, find the Jacobian matrix JF(r, θ): JF(r, θ) = [[∂(r cosθ)/∂r, ∂(r cosθ)/∂θ], [∂(r sinθ)/∂r, ∂(r sinθ)/∂θ]] JF(r, θ) = [[cosθ, -r sinθ], [sinθ, r cosθ]]

(b) Find F'(r, θ, φ) and JF(r, θ, φ) for F(r, θ, φ) = [r cosθ cosφ, r sinθ cosφ, r sinφ]

First, find the partial derivatives of F(r, θ, φ) with respect to r, θ, and φ: F'(r, θ, φ) = [∂(r cosθ cosφ)/∂r, ∂(r cosθ cosφ)/∂θ, ∂(r cosθ cosφ)/∂φ, ∂(r sinθ cosφ)/∂r, ∂(r sinθ cosφ)/∂θ, ∂(r sinθ cosφ)/∂φ, ∂(r sinφ)/∂r, ∂(r sinφ)/∂θ, ∂(r sinφ)/∂φ] F'(r, θ, φ) = [cosθ cosφ, -r sinθ cosφ, -r cosθ sinφ, sinθ cosφ, r cosθ cosφ, -r sinθ sinφ, sinφ, 0, r cosφ] Next, find the Jacobian matrix JF(r, θ, φ): JF(r, θ, φ) = [[∂(r cosθ cosφ)/∂r, ∂(r cosθ cosφ)/∂θ, ∂(r cosθ cosφ)/∂φ], [∂(r sinθ cosφ)/∂r, ∂(r sinθ cosφ)/∂θ, ∂(r sinθ cosφ)/∂φ], [∂(r sinφ)/∂r, ∂(r sinφ)/∂θ, ∂(r sinφ)/∂φ]] JF(r, θ, φ) = [[cosθ cosφ, -r sinθ cosφ, -r cosθ sinφ], [sinθ cosφ, r cosθ cosφ, -r sinθ sinφ], [sinφ, 0, r cosφ]]

(c) Find F'(r, θ, z) and JF(r, θ, z) for F(r, θ, z) = [r cosθ, r sinθ, z]

First, find the partial derivatives of F(r, θ, z) with respect to r, θ, and z: F'(r, θ, z) = [∂(r cosθ)/∂r, ∂(r cosθ)/∂θ, ∂(r cosθ)/∂z, ∂(r sinθ)/∂r, ∂(r sinθ)/∂θ, ∂(r sinθ)/∂z, ∂(z)/∂r, ∂(z)/∂θ, ∂(z)/∂z] F'(r, θ, z) = [cosθ, -r sinθ, 0, sinθ, r cosθ, 0, 0, 0, 1] Next, find the Jacobian matrix JF(r, θ, z): JF(r, θ, z) = [[∂(r cosθ)/∂r, ∂(r cosθ)/∂θ, ∂(r cosθ)/∂z], [∂(r sinθ)/∂r, ∂(r sinθ)/∂θ, ∂(r sinθ)/∂z], [∂(z)/∂r, ∂(z)/∂θ, ∂(z)/∂z]] JF(r, θ, z) = [[cosθ, -r sinθ, 0], [sinθ, r cosθ, 0], [0, 0, 1]]

Key concepts

Partial Derivatives

Partial derivatives are fundamental in multivariable calculus, as they represent how a function changes as each variable is varied independently. In simpler terms, when dealing with a function that has multiple inputs, a partial derivative tells you how much the output changes when you nudge just one of the inputs, keeping the others constant.

For instance, suppose we have a function that represents temperature as a function of both time and altitude. The partial derivative with respect to time would give us the rate at which the temperature is changing at a particular moment, ignoring changes in altitude.

In the original exercise, for a function \textbf{F}(r, \theta)\, partial derivatives were computed with respect to \r\ and \(\theta\). These calculations are crucial because they form the basis of the Jacobian matrix, which provides a comprehensive picture of how \textbf{F}\ changes in all directions at once. They help students understand the behavior of a multivariable function in its local neighborhood, which is extremely valuable for analysis in various fields such as physics, engineering, and economics.

Jacobian Matrix

The Jacobian matrix is a powerful tool in multivariable calculus that captures all the first-order partial derivatives of a vector-valued function. Think of it as a matrix-sized extension of the gradient for multivariable functions. It's like a snapshot that tells you how sensitive a multivariable output is to small changes in any of the inputs.

The Jacobian contains all the partial derivative information needed to approximate the function near a specific point, which is especially useful when functions become too complex for an exact analysis. It also has applications in fields like optimization, where one needs to find the maximum or minimum of function, or in dynamics, where it helps predict the behavior of a system over time.

In our exercise, the Jacobian matrices were calculated for different transformations of \( \mathbf{F} \), where each row of the Jacobian corresponds to the gradient of one component of the function with respect to the variables. This conveys how the entire function, not just one output, changes in response to variation in its inputs.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions of several variables. Unlike single-variable calculus, which deals with functions of one variable and their derivatives and integrals, multivariable calculus considers paths and surfaces in higher dimensions. This brings a new level of complexity and richness to the problems one can solve.

It plays a crucial role in understanding and modeling the physical world where many related quantities change together. Gradient, curl, and divergence are a few essential tools alongside partial derivatives and the Jacobian matrix. It provides the language and framework for describing rates of change and accumulations in multivariate settings and serves as a cornerstone for more advanced studies in mathematics and the applied sciences.

The exercises from the textbook are typical applications of multivariable calculus, involving transformations in two and three-dimensional spaces. The determination of partial derivatives and Jacobian matrices of these transformations enables a deeper insight into their geometric properties and how they alter the space they act upon.