Problem 415 - Calculus Volume 3

Question

Find the gradient. \(\quad f(x, y)=\frac{\sqrt{x}+y^{2}}{x y}\)

Step-by-Step Solution

Answer

The gradient is \( \nabla f = \left(-\frac{1}{2}x^{-3/2}y^{-1} - x^{-2}y, -x^{-1/2}y^{-2} + x^{-1}\right) \).

Step 1: Rewrite the Function

First, rewrite the given function to make it easier to differentiate. The function is initially given as \( f(x, y) = \frac{\sqrt{x} + y^2}{xy} \). This can be rewritten as \( f(x, y) = \frac{\sqrt{x}}{xy} + \frac{y^2}{xy} = \frac{x^{1/2}}{xy} + \frac{y}{x} \).

Step 2: Simplify the Function

Further simplify the rewritten function. It becomes \( f(x, y) = x^{-1/2}y^{-1} + x^{-1}y \).

Step 3: Partial Derivative with Respect to \(x\)

Find the partial derivative of \( f(x, y) \) with respect to \( x \). Differentiating, we get \( \frac{\partial f}{\partial x} = -\frac{1}{2}x^{-3/2}y^{-1} - x^{-2}y \).

Step 4: Partial Derivative with Respect to \(y\)

Find the partial derivative of \( f(x, y) \) with respect to \( y \). Differentiating, we obtain \( \frac{\partial f}{\partial y} = -x^{-1/2}y^{-2} + x^{-1} \).

Step 5: Express the Gradient

The gradient \( abla f \) is a vector consisting of the partial derivatives with respect to \( x \) and \( y \). So, \( abla f = \left(-\frac{1}{2}x^{-3/2}y^{-1} - x^{-2}y, -x^{-1/2}y^{-2} + x^{-1}\right) \).

Key Concepts

Partial Derivative

In the context of multivariable functions, partial derivatives serve as the building blocks for understanding how a function changes with respect to one variable while keeping the others constant. This concept is crucial, especially when dealing with functions of several variables. To compute a partial derivative, we differentiate the function with respect to one variable, treating the other variables as constants.

• For example, if you have a function like \( f(x, y) \), the partial derivative \( \frac{\partial f}{\partial x} \) measures the rate of change of the function with respect to \(x\), assuming \(y\) is constant.
• It gives us insight into the slope of the function along the \(x\) direction.
• Similarly, \( \frac{\partial f}{\partial y} \) measures the change as \(y\) varies.

These derivatives are fundamental when calculating gradients or working with complex surfaces and curves in multivariable calculus.

Differentiation

Differentiation is a mathematical process employed to find the derivative of a function. In simpler terms, it is the method used for assessing how a function's output value changes in response to changes in its input values. Differentiation can be understood more profoundly in the realm of single-variable and multivariable calculus.
In the context of single-variable functions, differentiation involves finding the rate at which a function changes at any given point. However, when extended to functions of multiple variables, such as \(f(x, y)\), differentiation becomes partial differentiation.

• This involves deriving the formula to find the slope of the function surface along the axes of the input variables.
• The differentiation of each variable treats the others as constants, simplifying the path to finding the solution.
• This approach helps in finding critical points, evaluating limits, or understanding the behavior of a system at a specific point.

Whether simple or complex, differentiation remains a cornerstone of mathematical analysis.

Vector Calculus

Vector calculus is a branch of mathematics focusing on vector fields and differential and integral operators. It enables handling directional changes, gradients, and transformations in multi-dimensional spaces. This concept becomes especially potent in physics and engineering, where phenomena in 3D space are analyzed.

• The gradient is a vital operator in vector calculus. It points in the direction of the greatest rate of increase of a function and its magnitude denotes the rate of increase.
• For example, in the exercise, the gradient \(abla f\) is represented as a vector combining the partial derivatives of the function with respect to each independent variable.
• Such operations assist in visualizing how a multivariate function behaves and evolves in its domain.

Vector calculus further manages vector fields, helping identify divergence, curl, and flux, which are essential in the fields of fluid dynamics, electromagnetism, and more.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of several variables. Contrary to functions of a single variable, these functions depend on multiple inputs, introducing greater complexity and depth.

• Key operations in multivariable calculus include finding partial derivatives, gradients, and more.
• This branch of calculus allows for the study of optimization, evaluating multiple factors simultaneously, which is particularly beneficial in fields like economics and physical sciences.
• The gradient vector derived in the exercise showcases how multiple partial derivatives can be compiled to understand the direction and rate of change across a multidimensional surface.

Understanding multivariable calculus introduces applications in real-world scenarios, enabling better modeling of natural and engineered systems.