Question

Within the cube \(\\{(x, y, z):|x| \leq 1\) \(|y| \leq 1,|z| \leq 1\\},\) where does div \(\mathbf{F}\) have the greatest magnitude when \(\mathbf{F}=\left\langle x^{2}-y^{2}, x y^{2} z, 2 x z\right\rangle ?\)

Short answer

The divergence of the vector field F has the greatest magnitude at the point (-1, 1, 1), where div F = -8.

Step-by-step solution

Calculate partial derivatives

Let's calculate the partial derivatives of each component of \(\mathbf{F}\) with respect to \(x\), \(y\), and \(z\): 1. \(F_x = x^2 - y^2 \implies \frac{\partial F_x}{\partial x} = 2x\) 2. \(F_y = x y^2 z \implies \frac{\partial F_y}{\partial y} = 2xy z\) 3. \(F_z = 2xz \implies \frac{\partial F_z}{\partial z} = 2x\)

Find the divergence of \(\mathbf{F}\)

To find the divergence of \(\mathbf{F}\), sum the partial derivatives we calculated in Step 1: \(\text{div} \ \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} = 2x + 2xy z + 2x\) Now, let's simplify this expression: \(\text{div} \ \mathbf{F} = 2x(1 + yz + 1)\) \(\text{div} \ \mathbf{F} = 4x(1 + yz)\)

Analyze the divergence expression for maximum magnitude

We want to find where the divergence of \(\mathbf{F}\) has the greatest magnitude within the cube \((-1 \leq x, y, z \leq 1)\). Notice that the magnitude of the divergence depends on the product \(yz\). This product will be maximized when \(y\) and \(z\) have the same sign and are as large as possible in magnitude. This occurs at the corners \((\pm 1, \pm 1, \pm 1)\), where \(y = z = \pm 1\). In that case, the divergence becomes: \(\text{div} \ \mathbf{F} = 4x(1 \pm 1) = 4x(2)\) or \(4x(-1)\) At the corners with \(y = z = 1\), the divergence is \(\text{div} \ \mathbf{F} = 8x\), and at the corners with \(y = z = -1\), the divergence is \(\text{div} \ \mathbf{F} = -4x\). The magnitude of the divergence will be greatest when \(x\) has the opposite sign of the product \(yz\). For \(y=z=1\), the maximum magnitude of \(\text{div} \ \mathbf{F}\) occurs when \(x=-1\) and \(\text{div} \ \mathbf{F} = -8\). For \(y=z=-1\), the maximum magnitude of \(\text{div} \ \mathbf{F}\) occurs when \(x=1\) and \(\text{div} \ \mathbf{F} = 4\). Therefore, the greatest magnitude of \(\text{div} \ \mathbf{F}\) occurs when \(x=-1\) and \(y=z=1\).

Confirm that the point is inside the cube

The point \((-1, 1, 1)\) lies within the cube since it satisfies the inequalities \(-1 \leq x, y, z \leq 1\). This means that the divergence of \(\mathbf{F}\) has the greatest magnitude at the point \((-1, 1, 1)\), where \(\text{div} \ \mathbf{F} = -8\).

Key concepts

Partial Derivatives

Partial derivatives are essential tools in calculus for understanding how a function changes as one of its variables changes while keeping other variables constant. In multivariable calculus, a function may depend on multiple variables, such as the vector field in our problem: \(\mathbf{F} = \langle x^{2}-y^{2}, xy^{2}z, 2xz \rangle\).
To explore how \(\mathbf{F}\) behaves, we compute its partial derivatives with respect to each variable. Each derivative measures the rate of change of one component of the vector field when only one variable is varied.

• For the component \(F_x = x^2 - y^2\), the partial derivative with respect to \(x\) is \(\frac{\partial F_x}{\partial x} = 2x\).
• For the component \(F_y = xy^2z\), with respect to \(y\), it is \(\frac{\partial F_y}{\partial y} = 2xyz\).
• For \(F_z = 2xz\), with respect to \(z\), it's \(\frac{\partial F_z}{\partial z} = 2x\).

By adding these partial derivatives, we gain insight into the behavior of the vector field \(\mathbf{F}\), laying the foundation for computing the divergence.

Vector Fields

Vector fields are mathematical constructions that assign a vector to every point in space. These vectors describe quantities having both magnitude and direction. In physics, field functions can represent various forces like gravity or electromagnetic forces.
In our exercise, the vector field \(\mathbf{F} = \langle x^{2}-y^{2}, xy^{2}z, 2xz \rangle\) represents a hypothetical field in a three-dimensional space delimited by a cube defined by \(|x|, |y|, |z| \leq 1\).

• The first component \(x^2 - y^2\) might affect how strongly the field moves along the \(x\) axis.
• The term \(xy^2z\) in the second component suggests interactions between \(x\), \(y\), and \(z\), affecting movements along the \(y\) axis.
• Similarly, \(2xz\) impacts the \(z\) axis direction.

Understanding such vector fields helps predict the intensity and direction of field-related interactions at any given point.

Maximum Magnitude

In the context of divergence, finding the maximum magnitude helps determine the point where the divergence is the greatest, indicating a significant field behavior, like the most intense outflow or inflow in that region.
To find this point, we first calculate the divergence of the vector field \(\mathbf{F}\), which is obtained by summing the partial derivatives of its components:\[\text{div} \ \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} = 2x + 2xyz + 2x = 4x(1 + yz)\]This expression reveals that the divergence depends on \(x\) and the product \(yz\).
By analyzing the conditions where \(yz\) is maximized, specifically at the cube's corners (such as \(y = z = \pm 1\)), we determine the divergence's maximum magnitude. For example, at \((-1, 1, 1)\), the maximum magnitude of \(\text{div} \ \mathbf{F}\) reaches \(-8\), crucial for understanding the vector field's behavior.

Multivariable Calculus

Multivariable Calculus extends single-variable calculus to functions of several variables. It is essential for modeling and analyzing complex systems like fluid dynamics, electromagnetics, and more.
In multivariable calculus:

• Functions may have forms such as \(f(x, y, z)\), assigning values based on multiple inputs.
• Techniques like partial derivatives measure change along each variable's axis independently.
• Divergence, gradients, and curl help understand fields and their behaviors in a multi-dimensional space.

This branch of calculus is crucial for dealing with real-world phenomena that inherently depend on multiple changing dimensions. Our problem with vector field \(\mathbf{F}\) in the cube is a classic example, illustrating how these concepts help predict where field properties like divergence peak in magnitude and influence.