Implicit function

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In mathematics, an implicit equation is a relation of the form <math>R(x_1, \dots, x_n) = 0,</math> where Template:Mvar is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is <math>x^2 + y^2 - 1 = 0.</math>

An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments.<ref name=Chiang>Template:Cite book</ref>Template:Rp For example, the equation <math>x^2 + y^2 - 1 = 0</math> of the unit circle defines Template:Mvar as an implicit function of Template:Mvar if Template:Math, and Template:Mvar is restricted to nonnegative values.

The implicit function theorem provides conditions under which some kinds of implicit equations define implicit functions, namely those that are obtained by equating to zero multivariable functions that are continuously differentiable.

Examples

Inverse functions

A common type of implicit function is an inverse function. Not all functions have a unique inverse function. If Template:Mvar is a function of Template:Mvar that has a unique inverse, then the inverse function of Template:Mvar, called Template:Math, is the unique function giving a solution of the equation

<math> y=g(x) </math>

for Template:Mvar in terms of Template:Mvar. This solution can then be written as

<math> x = g^{-1}(y) \,.</math>

Defining Template:Math as the inverse of Template:Mvar is an implicit definition. For some functions Template:Mvar, Template:Math can be written out explicitly as a closed-form expression — for instance, if Template:Math, then Template:Math. However, this is often not possible, or only by introducing a new notation (as in the product log example below).

Intuitively, an inverse function is obtained from Template:Mvar by interchanging the roles of the dependent and independent variables.

Example: The product log is an implicit function giving the solution for Template:Mvar of the equation Template:Math.

Algebraic functions

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} An algebraic function is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable Template:Mvar gives a solution for Template:Mvar of an equation

<math>a_n(x)y^n+a_{n-1}(x)y^{n-1}+\cdots+a_0(x)=0 \,,</math>

where the coefficients Template:Math are polynomial functions of Template:Mvar. This algebraic function can be written as the right side of the solution equation Template:Math. Written like this, Template:Mvar is a multi-valued implicit function.

Algebraic functions play an important role in mathematical analysis and algebraic geometry. A simple example of an algebraic function is given by the left side of the unit circle equation:

<math>x^2+y^2-1=0 \,. </math>

Solving for Template:Mvar gives an explicit solution:

<math>y=\pm\sqrt{1-x^2} \,. </math>

But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as Template:Math, where Template:Mvar is the multi-valued implicit function.

While explicit solutions can be found for equations that are quadratic, cubic, and quartic in Template:Mvar, the same is not in general true for quintic and higher degree equations, such as

<math> y^5 + 2y^4 -7y^3 + 3y^2 -6y - x = 0 \,. </math>

Nevertheless, one can still refer to the implicit solution Template:Math involving the multi-valued implicit function Template:Mvar.

Caveats

Not every equation Template:Math implies a graph of a single-valued function, the circle equation being one prominent example. Another example is an implicit function given by Template:Math where Template:Mvar is a cubic polynomial having a "hump" in its graph. Thus, for an implicit function to be a true (single-valued) function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a true function only after "zooming in" on some part of the Template:Mvar-axis and "cutting away" some unwanted function branches. Then an equation expressing Template:Mvar as an implicit function of the other variables can be written.

The defining equation Template:Math can also have other pathologies. For example, the equation Template:Math does not imply a function Template:Math giving solutions for Template:Mvar at all; it is a vertical line. In order to avoid a problem like this, various constraints are frequently imposed on the allowable sorts of equations or on the domain. The implicit function theorem provides a uniform way of handling these sorts of pathologies.

Implicit differentiation

In calculus, a method called implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.

To differentiate an implicit function Template:Math, defined by an equation Template:Math, it is not generally possible to solve it explicitly for Template:Mvar and then differentiate. Instead, one can totally differentiate Template:Math with respect to Template:Mvar and Template:Mvar and then solve the resulting linear equation for Template:Math to explicitly get the derivative in terms of Template:Mvar and Template:Mvar. Even when it is possible to explicitly solve the original equation, the formula resulting from total differentiation is, in general, much simpler and easier to use.

Examples

Example 1

Consider

<math>y + x + 5 = 0 \,.</math>

This equation is easy to solve for Template:Mvar, giving

<math>y = -x - 5 \,,</math>

where the right side is the explicit form of the function Template:Math. Differentiation then gives Template:Math.

Alternatively, one can totally differentiate the original equation:

<math>\begin{align}

\frac{dy}{dx} + \frac{dx}{dx} + \frac{d}{dx}(5) &= 0 \, ; \\[6px] \frac{dy}{dx} + 1 + 0 &= 0 \,. \end{align}</math>

Solving for Template:Math gives

<math>\frac{dy}{dx} = -1 \,,</math>

the same answer as obtained previously.

Example 2

An example of an implicit function for which implicit differentiation is easier than using explicit differentiation is the function Template:Math defined by the equation

<math> x^4 + 2y^2 = 8 \,.</math>

To differentiate this explicitly with respect to Template:Mvar, one has first to get

<math>y(x) = \pm\sqrt{\frac{8 - x^4}{2}} \,,</math>

and then differentiate this function. This creates two derivatives: one for Template:Math and another for Template:Math.

It is substantially easier to implicitly differentiate the original equation:

<math>4x^3 + 4y\frac{dy}{dx} = 0 \,,</math>

giving

<math>\frac{dy}{dx} = \frac{-4x^3}{4y} = -\frac{x^3}{y} \,.</math>

Example 3

Often, it is difficult or impossible to solve explicitly for Template:Mvar, and implicit differentiation is the only feasible method of differentiation. An example is the equation

<math>y^5-y=x \,.</math>

It is impossible to algebraically express Template:Mvar explicitly as a function of Template:Mvar, and therefore one cannot find Template:Math by explicit differentiation. Using the implicit method, Template:Math can be obtained by differentiating the equation to obtain

<math>5y^4\frac{dy}{dx} - \frac{dy}{dx} = \frac{dx}{dx} \,,</math>

where Template:Math. Factoring out Template:Math shows that

<math>\left(5y^4 - 1\right)\frac{dy}{dx} = 1 \,,</math>

which yields the result

<math>\frac{dy}{dx}=\frac{1}{5y^4-1} \,,</math>

which is defined for

<math>y \ne \pm\frac{1}{\sqrt[4]{5}} \quad \text{and} \quad y \ne \pm \frac{i}{\sqrt[4]{5}} \,.</math>

General formula for derivative of implicit function

If Template:Math, the derivative of the implicit function Template:Math is given by<ref name="Stewart1998">Template:Cite book</ref>Template:Rp

<math>\frac{dy}{dx} = -\frac{\,\frac{\partial R}{\partial x}\,}{\frac{\partial R}{\partial y}} = -\frac {R_x}{R_y} \,,</math>

where Template:Math and Template:Math indicate the partial derivatives of Template:Mvar with respect to Template:Mvar and Template:Mvar.

The above formula comes from using the generalized chain rule to obtain the total derivative — with respect to Template:Mvar — of both sides of Template:Math:

<math>\frac{\partial R}{\partial x} \frac{dx}{dx} + \frac{\partial R}{\partial y} \frac{dy}{dx} = 0 \,,</math>

hence

<math>\frac{\partial R}{\partial x} + \frac{\partial R}{\partial y} \frac{dy}{dx} =0 \,,</math>

which, when solved for Template:Math, gives the expression above.

Implicit function theorem

File:Implicit circle.svg

The unit circle can be defined implicitly as the set of points Template:Math satisfying Template:Math. Around point Template:Mvar, Template:Mvar can be expressed as an implicit function Template:Math. (Unlike in many cases, here this function can be made explicit as Template:Math.) No such function exists around point Template:Mvar, where the tangent space is vertical.

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Let Template:Math be a differentiable function of two variables, and Template:Math be a pair of real numbers such that Template:Math. If Template:Math, then Template:Math defines an implicit function that is differentiable in some small enough neighbourhood of Template:Open-open; in other words, there is a differentiable function Template:Mvar that is defined and differentiable in some neighbourhood of Template:Mvar, such that Template:Math for Template:Mvar in this neighbourhood.

The condition Template:Math means that Template:Math is a regular point of the implicit curve of implicit equation Template:Math where the tangent is not vertical.

In a less technical language, implicit functions exist and can be differentiated, if the curve has a non-vertical tangent.<ref name="Stewart1998"/>Template:Rp

In algebraic geometry

Consider a relation of the form Template:Math, where Template:Mvar is a multivariable polynomial. The set of the values of the variables that satisfy this relation is called an implicit curve if Template:Math and an implicit surface if Template:Math. The implicit equations are the basis of algebraic geometry, whose basic subjects of study are the simultaneous solutions of several implicit equations whose left-hand sides are polynomials. These sets of simultaneous solutions are called affine algebraic sets.

In differential equations

The solutions of differential equations generally appear expressed by an implicit function.<ref>Template:Cite book</ref>

Applications in economics

Marginal rate of substitution

In economics, when the level set Template:Math is an indifference curve for the quantities Template:Mvar and Template:Mvar consumed of two goods, the absolute value of the implicit derivative Template:Math is interpreted as the marginal rate of substitution of the two goods: how much more of Template:Mvar one must receive in order to be indifferent to a loss of one unit of Template:Mvar.

Marginal rate of technical substitution

Similarly, sometimes the level set Template:Math is an isoquant showing various combinations of utilized quantities Template:Mvar of labor and Template:Mvar of physical capital each of which would result in the production of the same given quantity of output of some good. In this case the absolute value of the implicit derivative Template:Math is interpreted as the marginal rate of technical substitution between the two factors of production: how much more capital the firm must use to produce the same amount of output with one less unit of labor.

Optimization

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Often in economic theory, some function such as a utility function or a profit function is to be maximized with respect to a choice vector Template:Mvar even though the objective function has not been restricted to any specific functional form. The implicit function theorem guarantees that the first-order conditions of the optimization define an implicit function for each element of the optimal vector Template:Math of the choice vector Template:Mvar. When profit is being maximized, typically the resulting implicit functions are the labor demand function and the supply functions of various goods. When utility is being maximized, typically the resulting implicit functions are the labor supply function and the demand functions for various goods.

Moreover, the influence of the problem's parameters on Template:Math — the partial derivatives of the implicit function — can be expressed as total derivatives of the system of first-order conditions found using total differentiation.

See also

• Implicit curve
• Functional equation
• Level set
  • Contour line
  • Isosurface
• Marginal rate of substitution
• Implicit function theorem
• Logarithmic differentiation
• Polygonizer
• Related rates
• Folium of Descartes

Further reading

• Template:Cite book
• Template:Cite book
• Template:Cite book

External links

• Archived at GhostarchiveTemplate:Cbignore and the Wayback MachineTemplate:Cbignore: {{#invoke:citation/CS1|citation

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