Question

Find the divergence of the vector field \(\mathbf{F}\) at the given point. $$ \begin{array}{lll} \text { Vector Field } & & \text { Point } \\ \mathbf{F}(x, y, z)=x^{2} z \mathbf{i}-2 x z \mathbf{j}+y z \mathbf{k} & & (2,-1,3) \\ \end{array} $$

Short answer

The divergence of the vector field \( \mathbf{F}(x, y, z) \) at the point (2, -1, 3) is 11.

Step-by-step solution

Find the Divergence of the Vector Field

The divergence of a vector field \( \mathbf{F}(x, y, z) \) is given by \( \nabla \cdot \mathbf{F} = \frac{\partial{F_{i}}}{\partial x} + \frac{\partial{F_{j}}}{\partial y} + \frac{\partial{F_{k}}}{\partial z} \). For the given vector field \( \mathbf{F}(x, y, z) = x^{2}z\mathbf{i} - 2xz\mathbf{j} + yz\mathbf{k} \), the partial derivative with respect to \( x \) for \( F_{i} = x^{2}z \) is \( \frac{\partial{F_{i}}}{\partial x} = 2xz \). The partial derivative with respect to \( y \) for \( F_{j} = - 2xz \) is \( \frac{\partial{F_{j}}}{\partial y} = 0 \). And the partial derivative with respect to \( z \) for \( F_{k} = yz \) is \( \frac{\partial{F_{k}}}{\partial z} = y \). So, the divergence of the vector field is \( \nabla \cdot \mathbf{F} = 2xz + 0 + y \).

Evaluate the Divergence at the Given Point

After finding the divergence of the vector field, the next task is to evaluate the divergence at the given point (2, -1, 3). All that is required is to substitute the coordinates of the given point into the divergence equation. So, by substituting \( x = 2, y = -1, z = 3 \) into \( \nabla \cdot \mathbf{F} = 2xz + 0 + y \), we get \( \nabla \cdot \mathbf{F}(2, -1, 3) = 2 * 2 * 3 + 0 + (-1) = 11. \)

Final Answer

The divergence of \( \mathbf{F}(x, y, z) = x^{2}z\mathbf{i} - 2xz\mathbf{j} + yz\mathbf{k} \) at the point (2,-1,3) is found to be 11.

Key concepts

Vector Calculus

Understanding how quantities vary in multivariable settings is the crux of vector calculus, an essential branch in mathematics often applied in physics and engineering. It comprises the study of vector fields, which are functions that assign vectors to points in space.

A vector field can represent various phenomena such as the flow of liquid or the intensity and direction of forces. For example, the vector field \( \mathbf{F}(x, y, z) = x^{2}z\mathbf{i} - 2xz\mathbf{j} + yz\mathbf{k} \) models a three-dimensional flow, where at each point in space, there's a vector with magnitude and direction determined by the coordinates of the point.

To investigate the behavior of such a field, we use the concept of divergence, which quantifies the degree to which the field is spreading out or converging at a certain point. Intuitively, if you imagine water flow represented by a vector field, regions with positive divergence are where water is being added (like a source), and regions with negative divergence where it's being removed (like a sink).

Partial Derivatives

To calculate the divergence of a vector field, we need to understand partial derivatives. In the realm of multivariable calculus, partial derivatives play the role akin to derivatives in single-variable calculus.

When we have a function that depends on multiple variables, a partial derivative measures how the function changes as one specific variable is varied, holding the other variables constant. In the exercise, when we compute \( \frac{\partial{F_{i}}}{\partial x} \) from the component \( F_{i} = x^{2}z \) of the vector field, we're finding out how this specific component changes with respect to \( x \) alone.

This process is replicated for each component of the vector field with respect to its corresponding variable, resulting in the expressions used to find the divergence. Thus, the ability to calculate these derivatives is fundamental for analyzing how the vector field behaves in the direction of each axis.

Multivariable Calculus

Multivariable calculus, as the name implies, involves calculus with more than one variable. This field extends the concepts of limit, derivative, and integral from functions of a single variable to those of multiple variables. Through this extension, it can handle more complex systems that are modeled by equations depending on various factors.

In the context of the given exercise, the use of multivariable calculus is evident. When we determine the divergence of the vector field \( \mathbf{F} \) at a certain point, we are using the tools provided by this discipline: evaluating partial derivatives, which are fundamental operations in multivariable calculus, and then applying them at a specific point in three-dimensional space. By doing so, we are essentially summarizing local properties of the vector field around that point in a single scalar quantity - the divergence.

This scalar gives us crucial information about the nature of the flow or field at that point, revealing whether the point acts like a source, with a positive divergence, or like a sink, with a negative divergence, or has no net flow, with a divergence of zero.