Question

In the implicit relationship \(F(x, y, z)=0,\) any two of the variables may be considered independent, which then determines the dependent variable. To avoid confusion, we may use a subscript to indicate which variable is held fixed in a derivative calculation; for example, \(\left(\frac{\partial z}{\partial x}\right)_{y}\) means that \(y\) is held fixed in taking the partial derivative of \(z\) with respect to \(x\). (In this context, the subscript does not mean a derivative.) a. Differentiate \(F(x, y, z)=0\) with respect to \(x,\) holding \(y\) fixed, to show that \(\left(\frac{\partial z}{\partial x}\right)_{y}=-\frac{F_{x}}{F_{z}}\) b. As in part (a), find \(\left(\frac{\partial y}{\partial z}\right)_{x}\) and \(\left(\frac{\partial x}{\partial y}\right)_{z_{2}}\) c. Show that \(\left(\frac{\partial z}{\partial x}\right)_{y}\left(\frac{\partial y}{\partial z}\right)_{x}\left(\frac{\partial x}{\partial y}\right)_{z}=-1\) d. Find the relationship analogous to part (c) for the case \(F(w, x, y, z)=0\).

Short answer

Answer: The relationship (∂z/∂x)_y(∂y/∂z)_x(∂x/∂y)_z for the function F(x, y, z) = 0 is equal to -1.

Step-by-step solution

Differentiate F(x, y, z) with respect to x holding y fixed

Differentiating F(x, y, z) with respect to x while keeping y fixed, we get: \[\frac{\partial}{\partial x}(F(x, y, z))= \frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial x}(\frac{\partial y}{\partial x})_{y}+\frac{\partial F_z}{\partial x}(\frac{\partial z}{\partial x})_{y} = 0\] Since y is held constant, \(\frac{\partial y}{\partial x}\)_y = 0. So the equation simplifies to: \[\frac{\partial F_x}{\partial x} + \frac{\partial F_z}{\partial x}(\frac{\partial z}{\partial x})_{y} = 0\]

Solve for (∂z/∂x)_y

To find the expression for (∂z/∂x)_y, isolate it in the equation: \[(\frac{\partial z}{\partial x})_{y} = -\frac{\partial F_x}{\partial x} \div \frac{\partial F_z}{\partial x} = -\frac{F_x}{F_z}\] Therefore: \[(\frac{\partial z}{\partial x})_{y} = -\frac{F_x}{F_z}\] #b. As in part (a), find (∂y/∂z)_x and (∂x/∂y)_z#

Differentiate F(x, y, z) with respect to z holding x fixed to find (∂y/∂z)_x

Differentiating F(x, y, z) with respect to z while keeping x constant, we get: \[\frac{\partial}{\partial z}(F(x, y, z)) = \frac{\partial F_x}{\partial z}(\frac{\partial x}{\partial z})_{x} + \frac{\partial F_y}{\partial z}(\frac{\partial y}{\partial z})_{x} + \frac{\partial F_z}{\partial z} = 0\] Since x is held constant, (\(\frac{\partial x}{\partial z}\)_x) = 0. So the equation simplifies to: \[\frac{\partial F_y}{\partial z}(\frac{\partial y}{\partial z})_{x} + \frac{\partial F_z}{\partial z} = 0\] Solving for (∂y/∂z)_x, we get: \[(\frac{\partial y}{\partial z})_{x} = -\frac{\partial F_z}{\partial z} \div \frac{\partial F_y}{\partial z} = -\frac{F_z}{F_y}\]

Differentiate F(x, y, z) with respect to y holding z constant to find (∂x/∂y)_z

Differentiating F(x, y, z) with respect to y while keeping z constant, we get: \[\frac{\partial}{\partial y}(F(x, y, z)) = \frac{\partial F_x}{\partial y}(\frac{\partial x}{\partial y})_{z} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial y}(\frac{\partial z}{\partial y})_{z} = 0\] Since z is held constant, (\(\frac{\partial z}{\partial y}\)_z) = 0. So the equation simplifies to: \[\frac{\partial F_x}{\partial y}(\frac{\partial x}{\partial y})_{z} + \frac{\partial F_y}{\partial y} = 0\] Solving for (∂x/∂y)_z, we get: \[(\frac{\partial x}{\partial y})_{z} = -\frac{\partial F_y}{\partial y} \div \frac{\partial F_x}{\partial y} = -\frac{F_y}{F_x}\] #c. Show that (∂z/∂x)_y(∂y/∂z)_x(∂x/∂y)_z = -1#

Multiply the expressions obtained for partial derivatives

Multiplying the expressions obtained in part a and part b, we have: \[(-\frac{F_x}{F_z})\cdot(-\frac{F_z}{F_y})\cdot(-\frac{F_y}{F_x})\]

Simplify the expression

Simplifying the above expression, we get: \[\frac{F_x}{F_z}\cdot\frac{F_z}{F_y}\cdot\frac{F_y}{F_x}=-1\] Thus, (∂z/∂x)_y(∂y/∂z)_x(∂x/∂y)_z = -1. #d. Find the relationship analogous to part (c) for the case F(w, x, y, z) = 0# For the function F(w, x, y, z) = 0, we have the following partial derivatives: 1. (∂x/∂y)_z = -Fx/Fy 2. (∂y/∂z)_w = -Fy/Fz 3. (∂z/∂w)_x = -Fz/Fw 4. (∂w/∂x)_y = -Fw/Fx The analogous relationship to part (c) will be a product of four partial derivatives. Let's multiply the above partial derivatives expressions: (-Fx/Fy)(-Fy/Fz)(-Fz/Fw)(-Fw/Fx) Simplifying the expression, we get: (Fx/Fy)(Fy/Fz)(Fz/Fw)(Fw/Fx) = 1 Thus, the relationship analogous to part (c) for the case F(w, x, y, z) = 0 is: (∂x/∂y)_z(∂y/∂z)_w(∂z/∂w)_x(∂w/∂x)_y = 1

Key concepts

Partial Derivatives

Partial derivatives are a type of derivative that measure how a function changes as one of its variables is varied, while the other variables are held constant. This is particularly important in multivariable calculus, where functions can depend on multiple variables. Understanding partial derivatives allows you to examine the rate of change with respect to one specific variable. This is especially useful in fields that work with models involving several linked factors, such as economics or physics.

When you calculate a partial derivative, you focus on one variable and treat the others as constants. For instance, if you have a function \(F(x, y, z)\), the partial derivative of \(F\) with respect to \(x\), written as \(\frac{\partial F}{\partial x}\), shows how \(F\) changes when only \(x\) changes.

• This process often uses subscripts to indicate which variables are held constant. For example, \(\left(\frac{\partial z}{\partial x}\right)_y\) implies finding the derivative with respect to \(x\) while keeping \(y\) constant.
• Partial derivatives are denoted with the curved 'd' symbol \(\partial\) instead of a regular 'd', which is used in total derivatives.

Chain Rule

The chain rule is a fundamental theorem in calculus used to compute the derivative of composite functions. This rule is particularly valuable when dealing with implicit differentiation and functions of multiple variables. It provides a way to differentiate a composite expression by breaking down the differentiation process into manageable steps.

In the context of multivariable calculus, where functions might depend on several interdependent variables, the chain rule helps simplify finding derivatives when inner functions themselves are functions of other variables.

• For instance, if you have a composite function \(F(g(h(x)))\), the chain rule allows you to find the derivative of \(F\) concerning \(x\) by evaluating: \(\frac{dF}{dh} \cdot \frac{dg}{dx}\).
• This methodology extends into higher dimensions, helping manage the complexity of functions with intertwined relationships among their variables.

This rule also plays a key role in implicit differentiation, allowing us to differentiate equations where the variable to differentiate is not isolated on one side of the equation.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus, such as differentiation and integration, from one variable to several variables. This field examines the behavior of functions that depend on multiple inputs, making it essential in understanding real-world phenomena that involve multiple influencing factors.

In multivariable calculus, functions are often expressed in terms of more than one variable, like \(F(x, y, z)\). This requires tools like partial derivatives and gradients to analyze their behavior. It provides insights into

• Optimization problems, where you might want to find maxima or minima of functions involving several variables.
• Changes and rates of change in systems like fluid dynamics, heat distribution, or population models in ecology.

Multivariable calculus also introduces new concepts like gradients and divergence, which have numerous applications in engineering, science, and economics.

Implicit Functions

Implicit functions are those not expressed directly as \(y = f(x)\) but rather involve a relationship among variables like \(x\), \(y\), and \(z\) together in an equation \(F(x, y, z) = 0\). These functions present challenges for differentiation as they don't isolate one variable fully in terms of the others.

Implicit differentiation is the technique used to find derivatives of such functions. Since the equation represents a relationship rather than a simple function mapping, differentiating them requires special strategies.

• Through implicit differentiation, you treat all terms of the equation with respect to each derivative's particular variable of interest.
• This approach often involves applying the chain rule and understanding partial derivatives.

One key property of implicit functions is they often enable expressing one variable as dependent on the others even when a simple expression is not present. This quality is particularly useful in physics or economics, where relationships are not always straightforwardly expressible in explicit function form.