Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume all partial derivatives exist. a. If \(z=(x+y) \sin x y,\) where \(x\) and \(y\) are functions of \(s,\) then \(\frac{\partial z}{\partial s}=\frac{d z}{d x} \frac{d x}{d s}\) b. Given that \(w=f(x(s, t), y(s, t), z(s, t)),\) the rate of change of \(w\) with respect to \(t\) is \(d w / d t\).

Short answer

Question: Determine whether the following statements are true or false, providing a counterexample or explanation if needed. a) If \(z = (x+y)\sin(xy)\) and \(x\) and \(y\) are functions of \(s\), then \(\frac{ dz }{ ds } = \frac{ dz }{ dx } \frac{ dx }{ ds }\) b) The rate of change of \(w = f(x(s, t), y(s, t), z(s, t))\) with respect to \(t\) is equal to \(\frac{ dw }{ dt }\). Answer: a) False. We found that \(\frac{\partial z}{\partial s}\) has the expression \(\left[(1 + y \cos(xy))\cdot(y + x\cos(xy))\right]\frac{dx}{ds} + \left[(1 + x \cos(xy))\cdot(x + y\cos(xy))\right]\frac{dy}{ds}\), which is not equal to \(\frac{d z}{d x} \frac{d x}{d s}\), providing a counterexample. b) True. The rate of change of \(w\) with respect to \(t\) is equal to \(\frac{d w}{d t}\), as shown by the expression derived using the chain rule: \(\frac{d w}{d t} = \frac{\partial w}{\partial x}\frac{d x}{d t} + \frac{\partial w}{\partial y}\frac{d y}{d t} + \frac{\partial w}{\partial z}\frac{d z}{d t}\).

Step-by-step solution

Statement a: Analysis

In this statement, we're given \(z = (x+y)\sin(xy)\), where \(x\) and \(y\) are functions of \(s\). To determine if this statement is true, we should find the partial derivative of \(z\) with respect to \(s\) and see if it equals to the claimed expression: \(\frac{dz}{dx}\frac{dx}{ds}\).

Statement a: Applying the Chain Rule

To find the partial derivative of \(z\) with respect to \(s\), we can consider \(z\) as a function of \(x(s)\) and \(y(s)\) (implying \(z = z\left(x(s), y(s)\right)\)) and then use the chain rule. According to the chain rule, we have: \(\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x}\frac{dx}{ds} + \frac{\partial z}{\partial y}\frac{dy}{ds}\)

Statement a: Evaluating the Partial Derivatives

Before we can provide a counterexample or explanation for statement a, we need to find the partial derivatives of \(z\) with respect to \(x\) and \(y\). They are computed as follows: - \(\frac{\partial z}{\partial x} = (1 + y \cos(xy))\cdot(y + x\cos(xy))\) - \(\frac{\partial z}{\partial y} = (1 + x \cos(xy))\cdot(x + y\cos(xy))\)

Statement a: The Partial Derivative of \(z\) w.r.t. \(s\)

Now we can plug the values of these partial derivatives of \(z\) into our expression for \(\frac{\partial z}{\partial s}\), which we obtained using the chain rule: \(\frac{\partial z}{\partial s} = \left[(1 + y \cos(xy))\cdot(y + x\cos(xy))\right]\frac{dx}{ds} + \left[(1 + x \cos(xy))\cdot(x + y\cos(xy))\right]\frac{dy}{ds}\) From this expression, it is clear that the statement is false as the partial derivative of \(z\) w.r.t. \(s\) does not equal to \(\frac{d z}{d x} \frac{d x}{d s}\). Therefore, we have provided a counterexample for statement a.

Statement b: Analysis

In this statement, we are asked if the rate of change of \(w = f(x(s, t), y(s, t), z(s, t))\) with respect to \(t\) is equal to \(\frac{dw}{dt}\). To check if this statement is true, we need to find the partial derivative of \(w\) with respect to \(t\) and see if it equals \(\frac{dw}{dt}\).

Statement b: Applying the Chain Rule

To find the partial derivative of \(w\) with respect to \(t\), we need to apply the chain rule, just like we did in statement a. According to the chain rule, we have: \(\frac{\partial w}{\partial t} = \frac{\partial w}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial w}{\partial y}\frac{\partial y}{\partial t} + \frac{\partial w}{\partial z}\frac{\partial z}{\partial t}\) In this expression, \(\frac{\partial t}{\partial t} = \frac{d t}{d t} = 1\). So, the given statement could also be rewritten as: \(\frac{d w}{d t} = \frac{\partial w}{\partial x}\frac{d x}{d t} + \frac{\partial w}{\partial y}\frac{d y}{d t} + \frac{\partial w}{\partial z}\frac{d z}{d t}\) This rewritten statement is exactly the expression we derived for the partial derivative of \(w\) with respect to \(t\). Therefore, statement b is true and it has been properly explained.

Key concepts

Partial Derivatives

Partial derivatives provide a way to look at how a function changes as we tweak just one of its input variables, keeping others constant. They are essential when working with functions of multiple variables, like our given function, \(z = (x + y) \sin(xy)\), with independent variables \(x\) and \(y\). If you think of a hill, the partial derivative with respect to \(x\) would tell you how steep the hill is in the \(x\) direction while ignoring any slope in the \(y\) direction.
To find these derivatives, we symbolically treat one variable as a constant and differentiate the function concerning the variable of interest. If we want the partial derivative with respect to \(x\), denoted as \(\frac{\partial z}{\partial x}\), we would calculate how \(z\) changes as \(x\) changes while holding \(y\) constant. Similarly, for \(y\), use \(\frac{\partial z}{\partial y}\).
In our exercise problem, this involves directly computing \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\), using standard derivative techniques but treating one variable as constant. This helps us significantly when applying the chain rule in scenarios where variables are functions themselves, such as \(x(s)\) and \(y(s)\).

Function of Multiple Variables

Functions of multiple variables refer to functions that depend on more than one input variable. In the given context, \(z\) is a function of \(x\) and \(y\), and both \(x\) and \(y\) are dependent on \(s\). This means the function depends indirectly on \(s\) through its two components, \(x(s)\) and \(y(s)\).
These types of functions are common in real-world applications where phenomena are influenced by several factors, such as temperature changes in a material depending on both time and space. In our exercise, \(z = (x+y) \sin(xy)\), the expression is a beautiful example of how multiple variables interact within a function.

• The advantage of understanding such functions is that it allows us to model complex behavior with multiple influences.
• It emphasizes the need for tools like the chain rule, which helps us understand changes in these functions when underlying variables change.
• When dealing with changes with respect to a particular variable, it becomes essential to take into account how every input variable contributes to the change.

Understanding this concept is crucial when analyzing scenarios involving indirect dependence, like dependency through other variables such as \(x(s)\).

Rate of Change

The rate of change of a variable tells us how one quantity changes in relation to another. In mathematics, this is often expressed as a derivative. The importance of understanding the rate of change lies in its ability to predict and explain real-world phenomena, like how quickly a car speeds up (acceleration as a rate of change of speed with time).
For the given function \(w = f(x(s, t), y(s, t), z(s, t))\), it is important to use the appropriate rules to determine how \(w\) changes concerning the variable \(t\). Here, the chain rule comes in handy yet again: it provides a structured way to dissect the influences of independent variables \(x\), \(y\), and \(z\) on \(w\), showing how they contribute to the change in \(w\) when \(t\) changes.
The statement \(\frac{dw}{dt} = \frac{\partial w}{\partial x} \frac{dx}{dt} + \frac{\partial w}{\partial y} \frac{dy}{dt} + \frac{\partial w}{\partial z} \frac{dz}{dt}\) confirms that the rate of change of \(w\) is indeed captured by considering the rates at which the individual components, \(x\), \(y\), and \(z\), are changing with \(t\).
Grasping the rate of change is not only pivotal for mathematical theory but also crucial for practical applications, particularly in physics, engineering, and economics, where understanding how things evolve with respect to time or other factors is essential.