Problem 32 - Advanced Engineering Mathematics

Question

Suppose \(u\) and \(v\) are the harmonic functions forming the real and imaginary parts of an analytic function. Show that the level curves \(u(x, y)=c_{1}\) and \(v(x, y)=c_{2}\) are orthogonal. [Hint: Consider the gradient of \(u\) and the gradient of \(v\). Ignore the case where a gradient vector is the zero vector.]

Step-by-Step Solution

Answer

The level curves \(u(x, y) = c_1\) and \(v(x, y) = c_2\) are orthogonal by the Cauchy-Riemann equations, making their gradients orthogonal.

Step 1: Understanding Harmonic Functions

A harmonic function is a twice continuously differentiable function that satisfies the Laplace equation, \(abla^2 u = 0\). In this case, both \(u\) and \(v\) are harmonic and form the real and imaginary parts of an analytic function. This implies that their gradients will play a significant role in understanding how the level curves behave.

Step 2: Calculating the Gradient of u

The gradient of \(u\), \(abla u\), is a vector \((u_x, u_y)\), where \(u_x\) and \(u_y\) are the partial derivatives of \(u\) with respect to \(x\) and \(y\), respectively. The gradient points in the direction of greatest increase of the function and is orthogonal to the level curves of \(u\).

Step 3: Calculating the Gradient of v

The gradient of \(v\), \(abla v\), is a vector \((v_x, v_y)\), where \(v_x\) and \(v_y\) are the partial derivatives of \(v\) with respect to \(x\) and \(y\), respectively. Similar to \(abla u\), this gradient indicates the direction of greatest change in \(v\) and is orthogonal to the level curves of \(v\).

Step 4: Cauchy-Riemann Equations

Since \(u\) and \(v\) are the real and imaginary parts of an analytic function, they satisfy the Cauchy-Riemann equations: \(u_x = v_y\) and \(u_y = -v_x\). These relationships allow us to connect the gradients of \(u\) and \(v\).

Step 5: Showing Orthogonality of Gradients

To show orthogonality, we compute the dot product \(abla u \cdot abla v = u_xv_x + u_yv_y\). Substituting the Cauchy-Riemann equations into this gives \(u_x(-u_y) + u_yu_x = 0\), proving that the gradients are orthogonal.

Step 6: Conclusion on Level Curves

Since the dot product of the gradients is zero, the gradients are orthogonal, and hence the level curves \(u(x, y) = c_1\) and \(v(x, y) = c_2\) are orthogonal wherever the gradients are non-zero.

Key Concepts

Cauchy-Riemann Equations

The Cauchy-Riemann equations are a set of conditions that must hold for a function to be analytic, meaning it has a complex derivative everywhere in its domain. When a complex function is expressed as a real part, \( u(x, y) \), and an imaginary part, \( v(x, y) \), these equations bind the partial derivatives of these components together.

These equations state that:

• \( u_x = v_y \)
• \( u_y = -v_x \)

This means the derivative of \( u \) with respect to \( x \) equals the derivative of \( v \) with respect to \( y \), while the derivative of \( u \) with respect to \( y \) is the negative of the derivative of \( v \) with respect to \( x \).

By connecting these derivatives, the equations ensure a harmonious relationship between \( u \) and \( v \). This is why they are crucial in proving that the level curves of these components are orthogonal.

Gradient

The gradient of a function is a vector that points in the direction of the greatest rate of increase of the function. For a function \( f(x, y) \), the gradient is expressed as \( abla f = (f_x, f_y) \), where \( f_x \) and \( f_y \) are the partial derivatives of \( f \) with respect to \( x \) and \( y \), respectively.

These vectors are not only indicative of direction but also serve an important role regarding the level curves, which are the paths where the function holds a constant value.

For harmonic functions, like \( u(x, y) \) and \( v(x, y) \), which form the real and imaginary parts of an analytic function, the gradients offer crucial insight. The vectors are orthogonal (at right angles) to the level curves of the functions they derive from. By understanding the orthogonality of these gradients, we can prove the orthogonality of the level curves for these harmonic functions.

Laplace Equation

The Laplace equation is a second-order partial differential equation given by \( abla^2 u = 0 \), where \( abla^2 \) indicates the Laplacian operator. In two dimensions, this equation can be expressed as:

• \( u_{xx} + u_{yy} = 0 \)

Where \( u_{xx} \) and \( u_{yy} \) are the second partial derivatives of \( u \) with respect to \( x \) and \( y \). A function satisfying this equation is termed harmonic.

The Laplace equation is a pivotal tool in mathematical physics and engineering, describing situations where there is an equilibrium or a steady state. For the functions \( u \) and \( v \), since they are harmonic, each satisfies the Laplace equation, ensuring a specific smoothness and balance in the behavior of \( u \) and \( v \).

This property of harmonic functions is essential for proving the orthogonality of their level curves, as the smooth and well-behaved nature of these functions, derived from the Laplace equation, facilitates the application of Cauchy-Riemann equations, which then leads to understanding curve orthogonality.