Problem 48 - Advanced Engineering Mathematics

Question

Assume that \(f\) and \(g\) are differentiable functions of two variables. Prove the given identity. $$ \nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}} $$

Step-by-Step Solution

Answer

Use the quotient rule on both partial derivatives; the gradient matches \(\frac{g \nabla f - f \nabla g}{g^2}\). Identity proven.

Step 1: Understand the Problem

We are given two differentiable functions, \(f(x, y)\) and \(g(x, y)\), and we need to prove the expression for the gradient of their quotient \( abla \left( \frac{f}{g} \right) \). The formula given is \( \frac{g abla f - f abla g}{g^2} \). We want to use differentiation techniques to arrive at this identity.

Step 2: Apply the Quotient Rule

The gradient of a function of two variables \( abla u = \left( \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y} \right) \). For the quotient \( u = \frac{f}{g} \), we use the quotient rule: \( \frac{\partial u}{\partial x} = \frac{g \frac{\partial f}{\partial x} - f \frac{\partial g}{\partial x}}{g^2} \), and similarly for the \( y \)-component: \( \frac{\partial u}{\partial y} = \frac{g \frac{\partial f}{\partial y} - f \frac{\partial g}{\partial y}}{g^2} \).

Step 3: Compute the Gradient Component-wise

First, calculate \( \frac{\partial f/g}{\partial x} \) using the quotient rule: \[ \frac{\partial}{\partial x} \left( \frac{f}{g} \right) = \frac{g \frac{\partial f}{\partial x} - f \frac{\partial g}{\partial x}}{g^2} \]. Then, calculate \( \frac{\partial f/g}{\partial y} \) similarly: \[ \frac{\partial}{\partial y} \left( \frac{f}{g} \right) = \frac{g \frac{\partial f}{\partial y} - f \frac{\partial g}{\partial y}}{g^2} \].

Step 4: Formulate the Gradient

The gradient is \( abla \left( \frac{f}{g} \right) = \left( \frac{g \frac{\partial f}{\partial x} - f \frac{\partial g}{\partial x}}{g^2}, \frac{g \frac{\partial f}{\partial y} - f \frac{\partial g}{\partial y}}{g^2} \right) \). Notice that this matches the identity formula \( \frac{g abla f - f abla g}{g^2} \), where \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \) and \( abla g = \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y} \right) \).

Step 5: Conclusion

We derived the gradient \( abla \left( \frac{f}{g} \right) \) using the quotient rule, and it matches the given formula: \( \frac{g abla f - f abla g}{g^2} \). Thus, the identity is proven correct.

Key Concepts

Differentiable Functions

A function is called differentiable if it has a derivative at every point in its domain. Differentiable functions are smooth and continuous, meaning there are no sudden changes or breaks. This is crucial for understanding gradients and derivatives.
In the context of multivariable calculus, a function, say \(f(x, y)\), is differentiable if you can compute the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\). These partial derivatives indicate how the function changes with respect to each variable individually.
Understanding differentiable functions helps us apply calculus techniques like gradient and chain rules to analyze more complex scenarios.

Quotient Rule

The quotient rule is an essential technique in calculus used to compute the derivative of the quotient of two differentiable functions. For a simple understanding: if you have a function \(u(x)=\frac{f(x)}{g(x)}\), where both \(f\) and \(g\) are differentiable, the derivative \(u'(x)\) is given by the formula:
\[ u'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{g(x)^2} \]
This rule ensures you get the correct rate of change of the quotient by using the rates of change of the numerator and denominator separately.
When dealing with functions of multiple variables, as in this exercise, the quotient rule extends to include partial derivatives, allowing us to calculate gradients for quotients like \(abla \left( \frac{f}{g} \right)\).

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus. They measure the rate of change of a function with respect to one of its multiple variables, keeping the other variables constant.
If \(f(x, y)\) is a function of two variables, the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) describe how \(f\) changes as \(x\) or \(y\) varies, respectively.
These derivatives help us understand the behavior of multivariable functions and are used to determine gradients, like in the exercise where they are applied in the quotient rule to compute \(abla\left(\frac{f}{g}\right)\).

Vector Calculus

Vector calculus is a field of mathematics that's vital for studying functions that depend on multiple variables. It introduces concepts like gradients, divergence, and curl, which are essential in physics and engineering.
The gradient, represented by \(abla\), gives a vector field indicating the direction of the steepest increase of a function at any point. This is particularly important in optimization and analytical settings.
With vector functions like \(\frac{f}{g}\), the gradient uses the techniques of vector calculus to combine partial derivatives into a comprehensive understanding of how the function behaves in all directions. The use of vector calculus simplifies complex interactions between variables, providing powerful tools for analysis and problem-solving.