Problem 258 - Calculus 3

Question

The \(x\) and \(y\) components of a fluid moving in two dimensions are given by the following functions: \(u(x, y)=2 y\) and \(v(x, y)=-2 x ; x \geq 0 ; y \geq 0 .\) The speed of the fluid at the point \((x, y)\) is \(s(x, y)=\sqrt{u(x, y)^{2}+v(x, y)^{2}} .\) Find \(\frac{\partial s}{\partial x}\) and \(\frac{\partial s}{\partial y}\) using the chain rule.

Step-by-Step Solution

Answer

\( \frac{\partial s}{\partial x} = \frac{2x}{\sqrt{x^2 + y^2}} \) and \( \frac{\partial s}{\partial y} = \frac{2y}{\sqrt{x^2 + y^2}} \).

Step 1: Determine Expressions for Speed Components

Given the speed equation \( s(x, y) = \sqrt{u(x, y)^2 + v(x, y)^2} \), we first need to express the components \( u(x, y) \) and \( v(x, y) \) in terms of \( x \) and \( y \). \( u(x, y) = 2y \) and \( v(x, y) = -2x \).

Step 2: Calculate Speed Function

Substitute the expressions for \( u \) and \( v \) into the speed function: \( s(x, y) = \sqrt{(2y)^2 + (-2x)^2} = \sqrt{4y^2 + 4x^2} = 2\sqrt{x^2 + y^2} \).

Step 3: Partial Derivative with Respect to x

To find \( \frac{\partial s}{\partial x} \), differentiate the speed equation \( s = 2\sqrt{x^2 + y^2} \) with respect to \( x \) using the chain rule: \[ \frac{\partial s}{\partial x} = \frac{d}{dx}(2\sqrt{x^2 + y^2}) = \frac{2}{2\sqrt{x^2 + y^2}} \cdot 2x = \frac{2x}{\sqrt{x^2 + y^2}} \].

Step 4: Partial Derivative with Respect to y

Similarly, to find \( \frac{\partial s}{\partial y} \), differentiate the speed function with respect to \( y \) using the chain rule: \[ \frac{\partial s}{\partial y} = \frac{d}{dy}(2\sqrt{x^2 + y^2}) = \frac{2}{2\sqrt{x^2 + y^2}} \cdot 2y = \frac{2y}{\sqrt{x^2 + y^2}} \].

Key Concepts

Chain Rule

In calculus, the chain rule is a fundamental principle used to find the derivative of a composite function. When dealing with multidimensional functions, especially in vector calculus, it becomes indispensable. If you have a function composed of two or more functions, you can use the chain rule to differentiate it with respect to a variable, even when the function is not explicitly dependent on that variable.
Using the chain rule helps to efficiently find partial derivatives, especially when a function is presented as a composition of other functions. In the given exercise, the speed function relies on both the components given; thus, using the chain rule allows us to compute the partial derivatives with respect to both variables easily.
To employ this, differentiate the outer function regarding an inner function, and then multiply with the derivative of the inner function concerning the variable of interest. This consistency in approach helps to simplify complex functions, turning something potentially overwhelming into manageable, smaller computations.

Vector Calculus

Vector calculus is the branch of mathematics that deals with differential and integral calculus of vector fields. In simpler terms, it helps us understand the behavior of vector fields, such as those describing fluid flow in space.
In fluid dynamics, the movement of fluid is often represented using vector fields where each vector describes a speed and direction of fluid flow at different points. This exercise introduces us to this concept by providing functions for the x and y components of a fluid's velocity in two dimensions.
Understanding vector fields requires analyzing components separately and then combining them. By doing this, we can predict the fluid flow's overall behavior. The exercise illustrates this by using the square root of the sum of the squares of the vector components to define fluid speed. This kind of vector operations leads us seamlessly into applying calculus, such as utilizing the chain rule for differentiation.

Fluid Dynamics

Fluid dynamics is the study of the flow of fluids, which can include liquids, gases, and in some cases, plasmas. It focuses on understanding patterns of movement within a fluid and the forces that cause them.
This exercise provides an avenue to explore simple fluid dynamics by observing how a fluid moves in a plane using defined x and y components of velocity. By determining how these components influence fluid speed, students get insight into two-dimensional flow patterns.
In practice, knowing how to calculate partial derivatives of speed with respect to each spatial variable helps in predicting how changes in shape or orientation affect fluid movement. Concepts learned here apply to understanding natural phenomena like ocean currents or designing engineering systems like pipeline flow. Overall, grasping these fundamental concepts in fluid dynamics can lead to better comprehension of a wide array of phenomena involving fluid behavior.