Problem 1 - A First Course in Continuum Mechanics

Question

Consider the scalar field \(\phi(x)=\left(x_{1}\right)^{2} x_{3}+x_{2}\left(x_{3}\right)^{2}\) and the vector field \(\boldsymbol{v}(\boldsymbol{x})=x_{3} \boldsymbol{e}_{1}+x_{2} \sin \left(x_{1}\right) \boldsymbol{e}_{3} .\) Find the components of \(\nabla \phi(\boldsymbol{x})\) \(\operatorname{and} \nabla \boldsymbol{v}(\boldsymbol{x})\).

Step-by-Step Solution

Answer

\(abla \phi(\boldsymbol{x}) = (2x_1x_3, (x_3)^2, (x_1)^2 + 2x_2x_3)\); Jacobian \(abla \boldsymbol{v}(\boldsymbol{x})\) is given by the matrix with the partial derivatives: \[\begin{bmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ x_2\cos(x_1) & \sin(x_1) & 0 \end{bmatrix}\].

Step 1: Find the gradient of the scalar field \(\phi(\boldsymbol{x})\)

To find \(abla \phi(\boldsymbol{x})\), apply the gradient operator to \(\phi(\boldsymbol{x})\) with respect to each variable. For a three-dimensional coordinate system, the gradient operator is \(abla = \left(\frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}, \frac{\partial}{\partial x_3}\right)\). Taking the partial derivatives, we get:\[abla \phi(\boldsymbol{x}) = \left(\frac{\partial \phi}{\partial x_1}, \frac{\partial \phi}{\partial x_2}, \frac{\partial \phi}{\partial x_3}\right)\]

Step 2: Calculate the component \(\frac{\partial \phi}{\partial x_1}\)

Compute the partial derivative of \(\phi\) with respect to \(x_1\):\[\frac{\partial \phi}{\partial x_1} = \frac{\partial}{\partial x_1}\left((x_1)^2x_3 + x_2(x_3)^2\right) = 2x_1x_3\]

Step 3: Calculate the component \(\frac{\partial \phi}{\partial x_2}\)

Compute the partial derivative of \(\phi\) with respect to \(x_2\):\[\frac{\partial \phi}{\partial x_2} = \frac{\partial}{\partial x_2}\left((x_1)^2x_3 + x_2(x_3)^2\right) = (x_3)^2\]

Step 4: Calculate the component \(\frac{\partial \phi}{\partial x_3}\)

Compute the partial derivative of \(\phi\) with respect to \(x_3\):\[\frac{\partial \phi}{\partial x_3} = \frac{\partial}{\partial x_3}\left((x_1)^2x_3 + x_2(x_3)^2\right) = (x_1)^2 + 2x_2x_3\]

Step 5: Find the gradient of the vector field \(\boldsymbol{v}(\boldsymbol{x})\)

To find \(abla \boldsymbol{v}(\boldsymbol{x})\), calculate the Jacobian matrix of \(\boldsymbol{v}\). For each component of \(\boldsymbol{v}\), \(v_i\), the corresponding row in \(abla \boldsymbol{v}\) is given by \(\left(\frac{\partial v_i}{\partial x_1}, \frac{\partial v_i}{\partial x_2}, \frac{\partial v_i}{\partial x_3}\right)\).

Step 6: Calculate the partial derivatives for the Jacobian of \(\boldsymbol{v}(\boldsymbol{x})\)

Find the partial derivatives of each component of \(\boldsymbol{v}\) with respect to each variable. The vector field is given by \(\boldsymbol{v}(\boldsymbol{x}) = x_3 \boldsymbol{e}_1 + x_2 \sin(x_1) \boldsymbol{e}_3\), thus we need to compute nine partial derivatives (three for each component) to form the Jacobian matrix.

Step 7: Construct the Jacobian matrix

Arrange the partial derivatives into the Jacobian matrix. The resulting matrix will be a 3x3 matrix where each entry \(\frac{\partial v_i}{\partial x_j}\) represents the partial derivative of the \(i\)-th component of \(\boldsymbol{v}\) with respect to the variable \(x_j\).

Key Concepts

Scalar Field Gradient

Understanding the gradient of a scalar field is crucial when studying vector calculus and physics. The gradient represents a vector pointing in the direction of the greatest rate of increase of the scalar field and its magnitude is the rate of increase in that direction.

For the scalar field \(\phi(x)=\left(x_{1}\right)^{2} x_{3}+x_{2}\left(x_{3}\right)^{2}\), the gradient, \(abla \phi(\boldsymbol{x})\), can be found by taking the partial derivatives with respect to each variable, \(x_1, x_2, x_3\). These derivatives provide the components of the gradient vector. In this case, the gradient is \(abla \phi(\boldsymbol{x}) = (2x_1x_3, (x_3)^2, (x_1)^2 + 2x_2x_3)\), meaning the scalar field increases the most rapidly in the direction of this vector.

Vector Field Jacobian

In vector calculus, the Jacobian matrix of a vector field represents how the field changes in different directions. Specifically, it contains all the first-order partial derivatives of the vector field components, revealing local linear approximations of the field.

For \(\boldsymbol{v}(\boldsymbol{x}) = x_{3} \boldsymbol{e}_{1}+x_{2} \sin\left(x_{1}\right) \boldsymbol{e}_{3}\), the Jacobian matrix \(abla \boldsymbol{v}(\boldsymbol{x})\) is a 3x3 matrix, where each element is a partial derivative \(\frac{\partial v_i}{\partial x_j}\). Calculating each partial derivative contributes to understanding how \(\boldsymbol{v}\) changes as any of the variables \(x_1, x_2\), or \(x_3\) are varied.

Partial Derivatives

Partial derivatives are fundamental in multivariable calculus. They measure how a function changes as one variable is varied, holding the other variables constant. Essentially, they extend the concept of a derivative to functions of several variables.

In the exercise, partial derivatives are used to determine the components of the gradient of \(\phi\) and the Jacobian matrix of \(\boldsymbol{v}\). The computation of \(\frac{\partial \phi}{\partial x_1}\), \(\frac{\partial \phi}{\partial x_2}\), and \(\frac{\partial \phi}{\partial x_3}\) are examples of how partial derivatives directly yield vital information about the rate of change of scalar and vector fields.

Continuum Mechanics Mathematics

The mathematics used in continuum mechanics involves the analysis of deformation, flow, and stresses within materials. It employs many mathematical concepts, including scalar and vector fields, gradients, and Jacobian matrices.

The gradient and the Jacobian from the earlier steps illustrate the types of mathematical tools used in continuum mechanics to describe the behavior of physical systems. By understanding the change of scalar quantities and how vector fields transform under various conditions, engineers and physicists can predict and analyze the physical phenomena observed in continuous media.