Problem 47 - Advanced Engineering Mathematics

Question

In Problems, assume that \(f\) and \(g\) are differentiable functions of two variables. Prove the given identity. $$ \nabla(f g)=f \nabla g+g \nabla f $$

Step-by-Step Solution

Answer

The given identity is proven by applying the product rule for derivatives to the gradient of the product \(f g\).

Step 1: Understand the Given Expression

The expression \( abla (f g) \) represents the gradient of the product of two functions \( f \) and \( g \). Our goal is to show that this gradient equals \( f abla g + g abla f \). The gradient, \( abla \), is a vector of partial derivatives.

Step 2: Recall the Gradient Formula

The gradient of a function \( h \) of two variables \( x \) and \( y \) is given by \( abla h = \left( \frac{\partial h}{\partial x}, \frac{\partial h}{\partial y} \right) \). We will apply this to the product \( f g \).

Step 3: Apply the Product Rule for Partial Derivatives

The derivative for products of functions follows the product rule. So, for \( \frac{\partial}{\partial x}(f g) \) we have:\[ \frac{\partial}{\partial x}(fg) = f \frac{\partial g}{\partial x} + g \frac{\partial f}{\partial x} \]This is because we applied the product rule individually to each partial derivative.

Step 4: Determine the Full Gradient

Similarly, applying the product rule to \( \frac{\partial}{\partial y}(f g) \) gives:\[ \frac{\partial}{\partial y}(fg) = f \frac{\partial g}{\partial y} + g \frac{\partial f}{\partial y} \]So the gradient is:\[ abla(fg) = \left( f \frac{\partial g}{\partial x} + g \frac{\partial f}{\partial x}, f \frac{\partial g}{\partial y} + g \frac{\partial f}{\partial y} \right) \]

Step 5: Connect to the Right-Hand Side of the Identity

The gradient \( abla g \) is \( \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y} \right) \) and the gradient \( abla f \) is \( \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \). Then:\[ f abla g = \left( f \frac{\partial g}{\partial x}, f \frac{\partial g}{\partial y} \right) \]\[ g abla f = \left( g \frac{\partial f}{\partial x}, g \frac{\partial f}{\partial y} \right) \]Adding these vectors gives us:\[ f abla g + g abla f = \left( f \frac{\partial g}{\partial x} + g \frac{\partial f}{\partial x}, f \frac{\partial g}{\partial y} + g \frac{\partial f}{\partial y} \right) \]

Step 6: Conclude the Proof

Since both sides of the identity \( abla(fg) = f abla g + g abla f \) result in the same vector, the proof is complete. This demonstrates that the gradient of a product of functions can be expressed as a sum of gradients, each scaled by the other function.

Key Concepts

Gradient

In vector calculus, the gradient is an essential concept that helps us understand how multi-variable functions change in space. The gradient of a function, denoted as \( abla \), provides a vector field containing the partial derivatives of the function with respect to its variables. This vector points in the direction of the greatest rate of increase of the function and its magnitude is the rate of increase in that direction.

For example, given a function \( f(x, y) \), its gradient is expressed as:

• \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \)

This means we are taking the derivative of \( f \) first with respect to \( x \), and then with respect to \( y \). If this gradient is applied to a product of functions, such as \( f \) and \( g \), it leads to a rule similar to the distributive property in algebra when calculating derivatives.

Product Rule

The product rule is a cornerstone of calculus that comes into play when differentiating products of two functions. When it comes to partial derivatives with multiple variables, this rule is very handy and often essential.

Let's say we want to differentiate the product of two functions, \( u(x, y) \) and \( v(x, y) \), with respect to \( x \). The product rule states:

• \( \frac{\partial}{\partial x}(uv) = u \frac{\partial v}{\partial x} + v \frac{\partial u}{\partial x} \)

It helps to visualize the product rule as distributing the differentiation to each factor of the product, while considering the constant participation of the other factor.

In the context of gradients, the product rule similarly extends to both variables when calculating both \( \partial / \partial x \) and \( \partial / \partial y \). This is why the formula for the gradient of a product \( abla(fg) = f abla g + g abla f \) holds true.

Partial Derivatives

A partial derivative is a derivative where we keep all but one of the variables constant. It captures the rate of change of a multi-variable function with respect to one variable, while maintaining the others as constants. This is a critical aspect of vector calculus, as most functions in this realm depend on multiple variables.

Here's an example for function \( f(x, y) \):

• \( \frac{\partial f}{\partial x} \) means we differentiate \( f \) considering \( x \) as a variable and \( y \) as a constant.
• \( \frac{\partial f}{\partial y} \) reverses the role, keeping \( x \) constant.

Partial derivatives are the building blocks of gradients. They enable the breakdown of changes in complex surfaces into simpler components, focusing on one direction at a time.
Working with partial derivatives is crucial when proving identities or theorems in vector calculus, as they form the backbone of every derivative operation when multiple variables are involved.