Implicit and explicit equations - Revision history

Khanh at 05:14, 14 November 2021

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Lila at 19:37, 18 October 2021

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Lila: Created page with "In mathematics, an implicit equation is a relation of the form {{math|1=''R''(''x''₁, …, ''x_n'') = 0}}, where {{mvar|R}} is a function of several varia..."

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Created page with "In mathematics, an implicit equation is a relation of the form {{math|1=''R''(''x''1, …, ''xn'') = 0}}, where {{mvar|R}} is a function of several varia..."

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In mathematics, an implicit equation is a relation of the form {{math|1=''R''(''x''1, …, ''xn'') = 0}}, where {{mvar|R}} is a function of several variables. For example, the implicit equation of the unit circle is {{math|1=''x''2 + ''y''2 − 1 = 0}}.

==Resources==
* [http://www.mathsmutt.co.uk/files/impex.htm Differentiating Implicit and Explicit Functions], Math Mutts
* [https://openstax.org/books/calculus-volume-1/pages/3-8-implicit-differentiation Implicit Differentiation], Openstax

← Older revision		Revision as of 19:37, 18 October 2021
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−	In most discussions of math, if the dependent variable y is a function of the independent variable x, we express y in terms of x. If this is the case, we say that y is an explicit function of x. For example, when we write the equation <math>y=x^2+1</math>, we are defining y explicitly in terms of x. On the other hand, if the relationship between the function y and the variable x is expressed by an equation where y is not expressed entirely in terms of x, we say that the equation defines y implicitly in terms of x. For example, the equation <math>~~y−x2~~=1</math> defines the function y=x2+1 implicitly.	+	In most discussions of math, if the dependent variable y is a function of the independent variable x, we express y in terms of x. If this is the case, we say that y is an explicit function of x. For example, when we write the equation <math>y=x^2+1</math>, we are defining y explicitly in terms of x. On the other hand, if the relationship between the function y and the variable x is expressed by an equation where y is not expressed entirely in terms of x, we say that the equation defines y implicitly in terms of x. For example, the equation <math>y-x^2=1</math> defines the function <math>y=x^2+1</math> implicitly.

	==Definition of Implicit Equation==		==Definition of Implicit Equation==

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 An '''implicit equation''' is a relation of the form {{math|1=''R''(''x''<sub>1</sub>, …, ''x<sub>n</sub>'') = 0}}, where {{mvar|R}} is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> − 1 = 0}}.
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 The implicit function theorem provides conditions under which some kinds of relations define an implicit function, namely relations defined as the indicator function of the zero set of some continuously differentiable multivariate function.
-==Examples==
+===Examples===
-===Inverse functions===
+====Inverse functions====
 A common type of implicit function is an inverse function. Not all functions have a unique inverse function. If {{mvar|g}} is a function of {{mvar|x}} that has a unique inverse, then the inverse function of {{mvar|g}}, called {{math|''g''<sup>−1</sup>}}, is the unique function giving a solution of the equation
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 '''Example:''' The product log is an implicit function giving the solution for {{mvar|x}} of the equation {{math|1=''y'' − ''xe''<sup>''x''</sup> = 0}}.
-===Algebraic functions===
+====Algebraic functions====
 An '''algebraic function''' is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable {{mvar|x}} gives a solution for {{mvar|y}} of an equation
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 ==Resources==
 * [https://en.wikipedia.org/wiki/Implicit_function Implicit Function], Wikipedia
 * [http://www.mathsmutt.co.uk/files/impex.htm Differentiating Implicit and Explicit Functions], Math Mutts
 * [https://openstax.org/books/calculus-volume-1/pages/3-8-implicit-differentiation Implicit Differentiation], Openstax

Implicit and explicit equations - Revision history

Khanh at 05:14, 14 November 2021

Lila at 19:37, 18 October 2021

Lila at 19:37, 18 October 2021

Lila at 19:33, 18 October 2021

Lila at 22:07, 17 September 2021

Lila at 22:07, 17 September 2021

Lila at 22:06, 17 September 2021

Lila: Created page with "In mathematics, an implicit equation is a relation of the form {{math|1=''R''(''x''1, …, ''xn'') = 0}}, where {{mvar|R}} is a function of several varia..."

Lila: Created page with "In mathematics, an implicit equation is a relation of the form {{math|1=''R''(''x''₁, …, ''x_n'') = 0}}, where {{mvar|R}} is a function of several varia..."