Problem 62 - Calculus: Early Transcendental Functions

Question

For \(\mathbf{v}=\left\langle x e^{x y}-1,2-y e^{x y}\right\rangle,\) show that \(\nabla \cdot \mathbf{v}=0\) and find a stream function \(g\).

Step-by-Step Solution

Answer

The divergence of the vector field is not zero, and as such, it contradicts the premise of the exercise. The stream function \(g\) is not obtainable under these circumstances.

Step 1: Compute the divergence of the vector

The divergence of a vector \(\mathbf{v}\) is given by \(\nabla \cdot \mathbf{v}=\frac{\partial v_{1}}{\partial x}+\frac{\partial v_{2}}{\partial y}\). Plug in the components of vector \(\mathbf{v}\) here to get \(\nabla \cdot \mathbf{v}= \frac{\partial(x e^{x y} - 1)}{\partial x} + \frac{\partial(2 - y e^{x y})}{\partial y}\) and simplify.

Step 2: Simplify the expression

Expand and simplify the expression. Now, \(\nabla \cdot \mathbf{v}= e^{x y} + x y e^{x y} - y e^{x y} + 0\). Notice that \(x y e^{x y} - y e^{x y} = 0\). So, \(\nabla \cdot \mathbf{v} = e^{x y}\) which is not equal to zero. So the question seems to have a mistake.

Step 3: Analysis

Given the calculation in step 2, it shows that the divergence of this vector field is not zero indicating there may be a mistake in the question. The second part of the problem can't be solved as the divergence is not zero. If it were, we could use the orthogonal gradient to find g but that is not the case here.

Key Concepts

Vector Field

A vector field is a construct in vector calculus that assigns a vector to every point in a given space. Imagine a map where at every point, an arrow is drawn. This arrow not only shows a direction but also a magnitude. That's what a vector field is all about. For example, the wind flow in the atmosphere or the magnetic field around a magnet can be expressed using vector fields.

In mathematical terms, a vector field is often represented as \(\mathbf{v}(x, y, z) = \langle v_1(x, y, z), v_2(x, y, z), v_3(x, y, z) \rangle\). Here, each component \(v_1, v_2,\) and \(v_3\) can vary depending on the coordinates \((x, y, z)\) in space.

• Vector field example: In our original exercise, we have \(\mathbf{v}=\left\langle x e^{x y}-1, 2-y e^{x y} \right\rangle\). This means at each point in 2D space \((x, y)\), the vector field assigns a vector that combines these component functions.
• Applications: Vector fields are used in physics to represent forces like gravity or electromagnetism. In engineering, they help model fluid flow.

Stream Function

A stream function serves a significant role in two-dimensional vector fields, particularly in incompressible fluid flows. When the divergence of a vector field is zero, it indicates that the vector field is divergence-free or solenoidal, hinting that there might exist a stream function.

This function, typically denoted by \(g(x, y)\), is related to the vector field components such that:

• \(\frac{\partial g}{\partial y} = v_1(x, y)\)
• \(-\frac{\partial g}{\partial x} = v_2(x, y)\)

Using these partial derivatives, you can find the stream function that fully characterizes the flow of the field without sources or sinks, ensuring conservation of mass.

In the given exercise, the task was to find a stream function \(g\), assuming \(abla \cdot \mathbf{v} = 0\), but since \(abla \cdot \mathbf{v} eq 0\), it was not possible to define such a function. This situation illustrates the critical condition of divergence-free fields necessary for stream functions.

Partial Derivatives

Partial derivatives are a fundamental part of calculus that allow us to study how functions change with respect to one variable while keeping other variables constant. This differs from ordinary derivatives where the change is considered with respect to only one variable.

Consider a function \(f(x, y)\). The partial derivative of \(f\) with respect to \(x\), denoted \(\frac{\partial f}{\partial x}\), measures the rate at which \(f\) changes as \(x\) changes, with \(y\) held constant. Similarly, \(\frac{\partial f}{\partial y}\) is the change in \(f\) with \(y\) changing while \(x\) is fixed.

• Why are they important? They help in understanding how multivariable functions behave in different dimensions. This is essential in fields like physics and engineering, where systems often depend on more than one variable.
• Role in divergence: In our exercise, calculating the divergence requires using partial derivatives of the vector field components, \(\frac{\partial v_1}{\partial x}\) and \(\frac{\partial v_2}{\partial y}\).

Partial derivatives give insight into the behavior and interaction of components in vector fields and are pivotal in fields such as optimization and machine learning where changes with respect to parameters need to be calculated. Understanding these concepts can significantly improve the approach to problem-solving in multivariable calculus.