Question

Assume that the vector field \(\mathbf{F}=\langle f, g\rangle\) is related to the stream function \(\psi\) by \(\psi_{y}=f\) and \(\psi_{x}=-g\) on a region \(R\). Prove that at all points of \(R\), the vector field is tangent to the streamlines (the level curves of the stream function).

Short answer

Question: Prove that the vector field \(\mathbf{F} = \langle f, g\rangle\), where \(\psi_{y} = f\) and \(\psi_{x} = -g\), is tangent to the streamlines (the level curves of the stream function) within the region \(R\). Answer: Since the tangent vector to the level curve \(\mathbf{t} = \langle f, g\rangle\) and the vector field \(\mathbf{F} = \langle f, g\rangle\) are equal at all points in the region \(R\), we can conclude that the vector field \(\mathbf{F}\) is tangent to the streamlines within the region \(R\).

Step-by-step solution

Determine the gradient of the stream function

The gradient of the stream function can be written as \(\nabla \psi = \langle \frac{\partial \psi}{\partial x}, \frac{\partial \psi}{\partial y}\rangle\). Given that \(\psi_{y} = f\) and \(\psi_{x} = -g\), the gradient is \(\nabla \psi = \langle -g, f\rangle\).

Define the level curve of the stream function

A level curve of a function is a curve in which the function remains constant. In this case, we consider the level curve of a stream function, which is given by \(L_c=\{ (x, y) \in R \mid \psi(x,y)=c \}\) for some constant \(c\). To find the tangent vector to this level curve, we need to compute the gradient of the stream function along this curve.

Tangent vector to the level curve

We want to find a tangent vector to the level curve \(L_c\) at a point \((x,y)\in R\). By definition, a tangent vector to a level curve is orthogonal to the gradient vector of the function. Hence, for two orthogonal vectors, the dot product is zero. Knowing that \(\nabla\psi = \langle -g, f\rangle\), the tangent vector to the level curve is given by \(\mathbf{t} = \langle f, g\rangle\). This is because the dot product between \(\mathbf{t}\) and \(\nabla\psi\) is \(( -g, f) \cdot (f, g) = (-g * f) + (f * g) = 0\), which indicates that they are orthogonal and thus, \(\mathbf{t}\) is a tangent vector to the level curve.

Compare the tangent vector and the vector field

Comparing the tangent vector to the level curve and the vector field, we have \(\mathbf{t} = \langle f, g\rangle\) and \(\mathbf{F} = \langle f, g\rangle\). Since they are equal at all points in the region \(R\), we can conclude that the vector field \(\mathbf{F}\) is tangent to the streamlines (the level curves of the stream function) within the region \(R\).

Key concepts

Gradient of Stream Function

In the realm of vector calculus, the gradient represents the multidimensional rate of change of a scalar field. For a stream function \(\psi\), which describes the flow of a fluid, the gradient is a pivotal concept. The gradient of \(\psi\) is denoted by \(abla \psi\) and is comprised of the partial derivatives of \(\psi\) with respect to the coordinates. In the exercise, \(abla \psi = \langle -g, f\rangle\) is derived by taking the partial derivatives \(\psi_{x} = -g\) and \(\psi_{y} = f\).

Understanding the gradient of a stream function is fundamental because it describes how the fluid's velocity changes in space. A unique property of the gradient of a stream function is that it is always perpendicular to the streamlines -- the paths that a massless fluid particle would follow in the flow. This characteristic ensures that streamlines are always contour lines of constant stream function value, leading us to the concept of level curves.

Level Curve of a Function

Imagine hiking on a hilly terrain and following a path that keeps you at the same altitude; that path is akin to the level curve of a function. In mathematics, a level curve is a curve along which a function retains a constant value. The exercise introduces the level curve \(L_c\) of the stream function \(\psi\), described as \(\{ (x, y) \in R \mid \psi(x,y)=c \}\), where \(c\) is the constant value of \(\psi\) on the curve.

These curves are instrumental in understanding the flow patterns of a fluid without calculating the actual movement. For students, it's essential to distinguish these curves as snapshots of the flow at specific values of \(\psi\), providing a visual representation of the entire flow field when taken together. The fact that the gradient vector at any point on a level curve is perpendicular to the curve itself allows us to gain insights into the flow's direction and behavior based on the orientation of these curves.

Tangent Vector Orthogonality

When dealing with curves in a field, a tangent vector at any point is a vector that 'touches' the curve only at that point, indicating the direction in which the curve is heading. For a level curve of a function, the tangent vector is crucial because it demonstrates the immediate path the curve takes. The principle of orthogonality, meaning at right angles, comes into play when one considers that the tangent vector to a level curve and the gradient of the function should be orthogonal to each other.

The exercise exemplifies this relationship by showing that the tangent vector \(\mathbf{t} = \langle f, g\rangle\) is orthogonal to the gradient \(abla \psi\) as their dot product results in zero. This orthogonality is a clear indication of perpendicularity and verifies that the direction of flow (tangent vector) is along the contour of equal stream function value (level curve). Comprehending this relationship equips students with the ability to analyze and predict flow patterns by just examining the geometry of level curves.

Vector Calculus

Vector calculus is an extension of calculus to vector fields, objects defining a direction and magnitude at points in space. It deals with operations and functions that arise in two and three dimensions, making it an indispensable tool for physicists and engineers. Through this branch of mathematics, concepts such as gradients, divergences, and curls -- which help explain the behavior of physical phenomena -- are defined.

The exercise presented is a brilliant example of applying vector calculus to analyze fluid flow. As students work through problems involving vector fields, like proving a vector field is tangent to streamlines, they employ vector calculus principles. One key takeaway is that understanding these principles allows for the translation of complex physical systems into mathematical terms that can be analyzed and solved, thus opening a gateway to modeling, understanding, and predicting physical realities in our everyday lives.