Question

Find the divergence of the vector field \(\mathrm{F}\). \(\mathbf{F}(x, y, z)=6 x^{2} \mathbf{i}-x y^{2} \mathbf{j}\)

Short answer

The divergence of the vector field \(\mathbf{F}\) is \( \nabla \cdot \mathbf{F} = 12x - 2xy\).

Step-by-step solution

Write down the vector components

First, identify and write down the components of the vector field. In this case, the vector field \(\mathbf{F}\) components are \(\mathbf{F_x} = 6x^2\), \(\mathbf{F_y} = -xy^2\), and \(\mathbf{F_z} = 0\), because no z component is given.

Compute the partial derivatives

Next, find the partial derivatives of each vector component with respect to its corresponding coordinate. So, calculate \( \frac{\partial F_x}{\partial x} \), \( \frac{\partial F_y}{\partial y} \), and \( \frac{\partial F_z}{\partial z} \). In this case, \( \frac{\partial F_x}{\partial x} = 12x \), \( \frac{\partial F_y}{\partial y} = -2xy \), and \( \frac{\partial F_z}{\partial z} = 0 \), as there is no z component.

Apply the formula for divergence

Finally, apply the divergence formula, \( \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \), to find the divergence of the vector field. Substituting the calcualted values into the formula, we get \( \nabla \cdot \mathbf{F} = 12x - 2xy + 0 \).

Key concepts

Partial Derivatives

In multivariable calculus, partial derivatives are a fundamental concept that embodies the rate at which a function changes with respect to one variable, while keeping other variables constant. Imagine you're hiking on a hilly terrain, the steepness of the hill in the direction you're facing represents a partial derivative.

For a function that depends on several variables, such as the vector field component functions in our exercise \( F_x = 6x^2 \) and \( F_y = -xy^2 \), calculating the partial derivatives \( \frac{\partial F_x}{\partial x} \) and \( \frac{\partial F_y}{\partial y} \) is analogous to finding how steep the hill is if you move either eastward or northward, while not wandering diagonally.

To calculate a partial derivative, you treat other variables as if they were constants and then differentiate normally with respect to the chosen variable. This process is crucial for understanding how a function behaves locally and is critical for concepts such as gradient, divergence, and curl in vector calculus.

Vector Calculus

Vector calculus is an extension of classical calculus to 2D and 3D space governed by vector fields. It's like having the superpower to describe wind flow patterns or magnetic fields, which are examples of vector fields. These fields describe a vector quantity (magnitude and direction) assigned at each point in space.

In our exercise, the vector field \( \mathbf{F} \) has two components given as \( 6x^2 \) and \( -xy^2 \) for \(\textbf{i}\) and \(\textbf{j}\) directions, respectively. Computing the divergence requires us to delve into the partial derivatives of these components to understand how the vector field behaves at each point. It's like deducing how a fluid might expand or compress as it flows through each point in space.

The steps detailed in the exercise lead you through computing the divergence of a vector field by using the partial derivatives of its component functions, giving a single scalar value as a result. The divergence in this context tells us about the rate at which the fluid represented by the vector field is diverging (spreading out) or converging (coming together) at a point.

Multivariable Calculus

Multivariable calculus is the branch of mathematics concerning calculus with more than one variable. Think of it as 3D modeling with math, where you need to consider multiple aspects simultaneously to understand the complete picture.

In such a setting, concepts like the divergence of a vector field represent an important tool. But before we get there, it's essential to begin with understanding how functions of multiple variables work, and that's where partial derivatives come in—as a first step towards building more complex constructions.

The exercise demonstrates an application of multivariable calculus through the process of finding a vector field's divergence, a concept that brings together both partial derivatives and vector calculus. By executing the steps as described, we synthesize the principles of partial derivatives and the geometric intuition of vector calculus to analyze and conclude something meaningful about the behavior of a vector field in space.