Problem 46 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The identity is proven by applying and summing gradients.

Step 1: Understanding the Gradient Operator

The gradient operator \( abla \) is a vector operator that acts on a scalar function to produce a vector. For a function \( f(x,y) \), it is defined as \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \). Similarly, for another function \( g(x,y) \), \( abla g = \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y} \right) \).

Step 2: Applying the Gradient to the Sum

When you apply the gradient operator to the sum \( f+g \), you're looking for \( abla (f+g) \). By definition, this means \( abla(f+g) = \left( \frac{\partial (f+g)}{\partial x}, \frac{\partial (f+g)}{\partial y} \right) \).

Step 3: Using the Sum of Derivatives Property

The derivative of a sum is the sum of the derivatives. Therefore, \( \frac{\partial (f+g)}{\partial x} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial x} \) and \( \frac{\partial (f+g)}{\partial y} = \frac{\partial f}{\partial y} + \frac{\partial g}{\partial y} \).

Step 4: Summing the Components

Combine these results to express the gradient of \( f+g \) as the sum of individual gradients: \( abla(f+g) = \left( \frac{\partial f}{\partial x} + \frac{\partial g}{\partial x}, \frac{\partial f}{\partial y} + \frac{\partial g}{\partial y} \right) \).

Step 5: Relating to the Original Identity

Notice that this is equivalent to \( abla f + abla g \), which is \( \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) + \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y} \right) = \left( \frac{\partial f}{\partial x} + \frac{\partial g}{\partial x}, \frac{\partial f}{\partial y} + \frac{\partial g}{\partial y} \right) \).

Step 6: Conclusion

Thus, \( abla(f+g) = abla f + abla g \), proving the identity as required.

Key Concepts

Gradient Operator

The gradient operator is a fundamental tool in multivariable calculus. It is represented by the symbol \( abla \) (nabla) and serves to transform a scalar function into a vector.
For any scalar function, \( f(x, y) \), the gradient is a vector comprised of its partial derivatives: \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \). This vector indicates the direction of steepest increase of the function, with its magnitude giving the rate of increase.
The concept is essential for applications including optimization and finding normal directions to surfaces, making it a key element in the study of multivariable functions.

Sum of Derivatives

In calculus, understanding how derivatives behave when functions are combined is crucial. The sum of derivatives property states that the derivative of a sum is the sum of the derivatives.
Mathematically, for functions \( f \) and \( g \), this translates to: \( \frac{\partial (f+g)}{\partial x} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial x} \). Similarly, \( \frac{\partial (f+g)}{\partial y} = \frac{\partial f}{\partial y} + \frac{\partial g}{\partial y} \).
By applying this property, we can easily extend the understanding of how the gradient operator acts on the sum of functions, thus confirming the original identity \( abla(f+g) = abla f + abla g \).

Differentiable Functions

Differentiable functions are functions that have derivatives. They are fundamental in analysis because they indicate smooth and continuous behavior, allowing calculus techniques to be applied.
If a function \( f(x, y) \) is differentiable, it means that partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) exist, forming the gradient vector. Differentiability implies that at any given point, the function can be well-approximated by a linear function (a plane in the case of two variables).
This property not only underpins the calculations you'll perform but also ensures that any manipulations involving derivatives, such as sums, can be carried out reliably. Differentiable functions create the framework where calculus's powerful tools can be leveraged.

Multivariable Calculus

Multivariable calculus is a branch of calculus that deals with functions of multiple variables. It extends the concepts of single-variable calculus, such as differentiation and integration, to functions of two or more variables.
In this expanded view, you explore the behavior of functions that depend on multiple inputs, and the gradient operator is an exemplary tool for this purpose. Concepts like partial derivatives and gradient vectors help us analyze and visualize the multi-dimensional changes of these functions.
Multivariable calculus plays a vital role in fields such as engineering, physics, and economics by providing the mathematical framework needed to model and solve real-world problems involving several changing factors.