Problem 1 - Advanced Engineering Mathematics

Question

In Problems, compute the gradient for the given function. $$ f(x, y)=x^{2}-x^{3} y^{2}+y^{4} $$

Step-by-Step Solution

Answer

The gradient is \( \nabla f = (2x - 3x^2 y^2, -2x^3 y + 4y^3) \).

Step 1: Understand the Function

We are given the function \( f(x, y) = x^2 - x^3 y^2 + y^4 \). To find the gradient \( abla f \), we need to compute the partial derivatives of the function with respect to its variables, \( x \) and \( y \).

Step 2: Compute Partial Derivative with Respect to x

The partial derivative of \( f \) with respect to \( x \) is obtained by differentiating \( f(x, y) = x^2 - x^3 y^2 + y^4 \) while treating \( y \) as a constant.\[ f_x = \frac{\partial}{\partial x}(x^2) - \frac{\partial}{\partial x}(x^3y^2) + \frac{\partial}{\partial x}(y^4) \]\[ f_x = 2x - 3x^2 y^2 + 0 \]So, \( f_x = 2x - 3x^2 y^2 \).

Step 3: Compute Partial Derivative with Respect to y

The partial derivative of \( f \) with respect to \( y \) is done by differentiating \( f(x, y) = x^2 - x^3 y^2 + y^4 \) while treating \( x \) as a constant.\[ f_y = \frac{\partial}{\partial y}(x^2) - \frac{\partial}{\partial y}(x^3y^2) + \frac{\partial}{\partial y}(y^4) \]\[ f_y = 0 - 2x^3 y + 4y^3 \]Therefore, \( f_y = -2x^3 y + 4y^3 \).

Step 4: Write the Gradient

The gradient of \( f \), denoted as \( abla f \), is a vector composed of the partial derivatives \( f_x \) and \( f_y \). Thus,\[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \]\[ abla f = (2x - 3x^2 y^2, -2x^3 y + 4y^3) \].

Key Concepts

Partial Derivatives

Partial derivatives are an essential concept in calculus, especially when dealing with functions of multiple variables. They tell us how a function changes as we vary one of the variables while keeping the others constant. For example, in the function \( f(x, y) = x^2 - x^3 y^2 + y^4 \), the partial derivative of \( f \) with respect to \( x \) is found by varying \( x \) alone and treating \( y \) as a constant.

• The symbol \( \frac{\partial}{\partial x} \) represents the derivative with respect to \( x \).
• Solving \( \frac{\partial}{\partial x}(x^2 - x^3 y^2 + y^4) \) yields \( f_x = 2x - 3x^2 y^2 \).
• The partial derivative with respect to \( y \), represented as \( f_y \), is similarly computed by varying \( y \) while keeping \( x \) constant, resulting in \( f_y = -2x^3 y + 4y^3 \).

Partial derivatives offer a way to look at changes along one dimension at a time, enabling us to dissect complex multivariable functions.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions with two or more variables. This branch of calculus allows us to explore how functions behave in higher dimensions.
For the function \( f(x, y) = x^2 - x^3 y^2 + y^4 \), we can think about it as a surface in three-dimensional space.

• Unlike single-variable calculus, which deals with lines and curves, multivariable calculus deals with surfaces and hypersurfaces.
• The gradient \( abla f \), a vector of partial derivatives, provides a way to examine the function's rate of change in any direction.
• The choice of variables—like \( x \) and \( y \) in our example—defines the dimensions along which we're analyzing the function.

Multivariable calculus is beneficial for understanding phenomena in physics, engineering, and even in economic models, where multiple factors affect the outcome.

Vector Calculus

Vector calculus is a branch of mathematics that focuses on differentiating and integrating vector fields, typically in two or three dimensions.
When dealing with functions like \( f(x, y) = x^2 - x^3 y^2 + y^4 \), these functions can be represented as vector fields.

• The gradient \( abla f = (f_x, f_y) \) is a vector field representing the direction and rate of fastest increase of the function.
• The length or magnitude of this gradient vector gives us the steepness of the incline at any point on the surface described by the function.
• Vector calculus tools allow us to calculate important properties of the field, such as divergence and curl, which are used in physics to describe fluid flow, electromagnetism, and more.

Understanding vector fields and operations is key to modeling real-world scenarios where directional derivatives and field lines play crucial roles.