Problem 61 - Calculus: Early Transcendental Functions

Question

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

Step-by-Step Solution

Answer

The divergence of the given vector field \(v\) is zero and the stream function \(g\) for this field is \(g = xy^2 - y^2x + \frac{x^2}{2} + cx\), where c is a constant.

Step 1: Calculate the Divergence

The divergence of a two-dimensional vector field \(v = (P, Q)\) is defined as \(\nabla \cdot v = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}\). In this case, P is \(2xy\) and Q is \(-y^2 + x\). By partial differentiation, we find that the derivatives are \(\frac{\partial P}{\partial x} = 2y\) and \(\frac{\partial Q}{\partial y} = -2y\). Taking the sum of these gives us \(\nabla \cdot v = 2y - 2y = 0\). Thus, the divergence of \(v\) is zero, as required.

Step 2: Identify the Stream Function

A stream function \(g\) for a two-dimensional vector field \(v = (P, Q)\) is a function such that \(P = \frac{\partial g}{\partial y}\) and \(Q = -\frac{\partial g}{\partial x}\). This means we have to solve the two equations \(2xy = \frac{\partial g}{\partial y}\) and \(-y^2 + x = -\frac{\partial g}{\partial x}\) for \(g\).

Step 3: Solve for the Stream Function

Starting with the first equation, we can integrate \(2xy = \frac{\partial g}{\partial y}\) with respect to \(y\) to give \(g = xy^2 + h(x)\), where \(h(x)\) is a function of \(x\) determined by the second equation. Next, differentiate \(g\) with respect to \(x\) and set the result equal to the right-hand side of the second equation. This results in \(y^2 + h'(x) = -y^2 + x\). Solving for \(h'(x)\), we see that \(h'(x) = -2y^2 + x\). Therefore the function \(h(x)\) would be \(-y^2x + x\). Thus, the stream function is \(g = xy^2 - y^2x + \frac{x^2}{2} + cx\), where c is a constant.

Key Concepts

Vector Field Divergence

The divergence of a vector field is a crucial concept in vector calculus that measures the density of the outward flux of a vector field from an infinitesimal volume around a given point. In simple terms, it tells us how much the vector field spreads out or converges at a particular point.

For a two-dimensional vector field represented by \(v = (P, Q)\), the divergence is given by the formula \(abla \cdot v = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}\). The process involves partial differentiation of the vector field's components with respect to their respective variables. If the divergence is zero, as in the given exercise where \(abla \cdot v = 0\), it indicates that the vector field is incompressible or has no net 'source' or 'sink' at that point. This attribute can have significant implications in fields such as fluid dynamics and electromagnetism.

In the solved exercise, by computing the partial derivatives of the vector field components P and Q, we ensure that the vector field divergence equates to zero, confirming a key characteristic of the field under consideration.

Partial Differentiation

Partial differentiation is the process of differentiating a function of several variables with respect to one variable while keeping the other variables constant. It is akin to taking the derivative of a function in single-variable calculus, but with the additional complexity of navigating a multi-dimensional function landscape.

The partial derivatives in the context of a two-dimensional vector field \(v = (P, Q)\) represent the rate of change of the vector components \(P\) and \(Q\) with respect to the variables \(x\) and \(y\), respectively. This process is foundational for finding the divergence of a vector field, as seen in the exercise with \(\frac{\partial P}{\partial x}\) and \(\frac{\partial Q}{\partial y}\). These derivatives are then added to find the divergence, which is a vital step to understanding the behavior of the vector field at a point.

The exercise demonstrates the practical application of partial differentiation to verify the divergence of a given vector field by methodically evaluating the derivatives of its components.

Stream Function Calculus

The concept of a stream function is a valuable tool used in fluid dynamics to describe the flow of incompressible two-dimensional fluids. In mathematics, particularly vector calculus, it acts as a potential function for a vector field, which allows us to visualize the flow lines.

The stream function \(g\) is related to the vector field \(v = (P, Q)\) such that \(P = \frac{\partial g}{\partial y}\) and \(Q = -\frac{\partial g}{\partial x}\), as established in the exercise. By solving these partial differential equations, we can find a function that encapsulates the essence of the field's flow pattern.

In the given problem, following the determination of the stream function involves integration and differentiation - key operations in stream function calculus. The resulting function \(g\), which ultimately accounts for the vector field's flow characteristics, is found through meticulous calculation. This stream function provides insight into the potential behavior of particles within the vector field, which is of high relevance in physical applications such as the dynamics of fluid flow.