Question

Explain how to compute the divergence of the vector field \(\mathbf{F}=\langle f, g, h\rangle\).

Short answer

Question: Calculate the divergence of the vector field \(\mathbf{F} = \langle xy, yz, xz \rangle\). Answer: To calculate the divergence of the given vector field, follow the steps outlined in the solution. In this case, the components of the vector field are \(f = xy\), \(g = yz\), and \(h = xz\). Step 2: Compute the partial derivatives 1. \(\frac{\partial f}{\partial x} = y\) 2. \(\frac{\partial g}{\partial y} = z\) 3. \(\frac{\partial h}{\partial z} = x\) Step 3: Sum up the partial derivatives \(\nabla \cdot\mathbf{F} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} + \frac{\partial h}{\partial z} = y + z + x\) Step 4: Simplify the result The divergence of the given vector field \(\mathbf{F}\) is \(\nabla \cdot\mathbf{F} = x + y + z\).

Step-by-step solution

Identify the components of the vector field

The given vector field is \(\mathbf{F} = \langle f, g, h\rangle\). The components of the vector field are \(f\), \(g\), and \(h\), which represent scalar functions of possibly multiple variables.

Compute the partial derivatives

To find the divergence, we must compute the partial derivatives of the components with respect to their respective variables. Let's assume the vector field is a function of \(x\), \(y\), and \(z\). This means we will find the following partial derivatives: 1. \(\frac{\partial f}{\partial x}\) - the partial derivative of \(f\) with respect to \(x\) 2. \(\frac{\partial g}{\partial y}\) - the partial derivative of \(g\) with respect to \(y\) 3. \(\frac{\partial h}{\partial z}\) - the partial derivative of \(h\) with respect to \(z\)

Sum up the partial derivatives

After computing the three partial derivatives, we will add them up to find the divergence of the vector field \(\mathbf{F}\). The divergence, denoted as \(\nabla \cdot\mathbf{F}\) or \(\text{div}(\mathbf{F})\), is given by the following expression: $$\nabla \cdot\mathbf{F} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} + \frac{\partial h}{\partial z}$$

Simplify the result

If possible, simplify the resulting expression obtained from step 3. The final expression we get is the divergence of the given vector field \(\mathbf{F}\).

Key concepts

Vector Calculus

Vector calculus is a crucial branch of mathematics that focuses on vector quantities and incorporates operations such as differentiation and integration. It plays a significant role in understanding physical phenomena like fluid flow, electromagnetic fields, and gravitational forces.
Key operations in vector calculus include:

• Gradient, which measures the rate of increase of a scalar field.
• Divergence, which quantifies a vector field's tendency to originate from or converge into a point.
• Curl, which assesses the rotation or swirling strength of a vector field.

Understanding these tools allows one to describe and predict the behavior of vector fields, making vector calculus an essential tool in physics, engineering, and computer graphics. By learning vector calculus, students can better grasp the dynamics of various physical systems.

Partial Derivatives

Partial derivatives are a fundamental concept in calculus used when dealing with functions of multiple variables. They represent the rate of change of a function concerning one variable while holding other variables constant.
When calculating the divergence of a vector field, partial derivatives are used to examine how each component of the vector field changes with respect to each of its variables. For a vector field \(\mathbf{F} = \langle f(x, y, z), g(x, y, z), h(x, y, z)\rangle\), this involves:

• Finding \(\frac{\partial f}{\partial x}\)
• Finding \(\frac{\partial g}{\partial y}\)
• Finding \(\frac{\partial h}{\partial z}\)

The sum of these partial derivatives gives us the divergence \(abla \cdot \mathbf{F}\). Trust in partial derivatives is key as they greatly assist in visualizing and understanding how multi-variable functions behave and interrelate.

Vector Fields

Vector fields are mathematical constructions where a vector is assigned to every point in a specific space. These fields represent both magnitude and direction, and they are essential in describing physical contexts like the flow of fluids or the force exerted by a gravitational field.
In a typical vector field depiction, vectors can represent quantities such as wind speed, magnetic force, or fluid velocity. A vector field is expressed as \(\mathbf{F} = \langle f(x, y, z), g(x, y, z), h(x, y, z)\rangle\), where each component function expresses a different facet of the field's behavior.
To compute the divergence of such a field, as shown in vector calculus, determine how these vectors spread out from a point. By computing and understanding vector fields and their divergence, one can predict systems' behaviors in diverse scientific and engineering scenarios.