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Implicit function theorem

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In multivariable calculus , the implicit function theorem is a tool that allows relations to be converted to functions of several real variables . It does so by representing the relation as the graph of a function . There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.

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100-414: More precisely, given a system of m equations f i   ( x 1 , ..., x n , y 1 , ..., y m ) = 0, i = 1, ..., m (often abbreviated into F ( x , y ) = 0 ), the theorem states that, under a mild condition on the partial derivatives (with respect to each y i ) at a point, the m variables y i are differentiable functions of the x j in some neighborhood of

200-604: A g {\displaystyle g} that works near the point ( a , b ) {\displaystyle ({\textbf {a}},{\textbf {b}})} . In other words, we want an open set U ⊂ R n {\displaystyle U\subset \mathbb {R} ^{n}} containing a {\displaystyle {\textbf {a}}} , an open set V ⊂ R m {\displaystyle V\subset \mathbb {R} ^{m}} containing b {\displaystyle {\textbf {b}}} , and

300-462: A , b ) {\displaystyle ({\textbf {a}},{\textbf {b}})} , the Jacobian matrix is ( D f ) ( a , b ) = [ ∂ f 1 ∂ x 1 ( a , b ) ⋯ ∂ f 1 ∂ x n (

400-392: A {\textstyle x=a} when Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition. A function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } , that is complex-differentiable at a point x = a {\textstyle x=a}

500-483: A ) = b {\displaystyle g(\mathbf {a} )=\mathbf {b} } , and f ( x , g ( x ) ) = 0   for all   x ∈ U {\displaystyle f(\mathbf {x} ,g(\mathbf {x} ))=\mathbf {0} ~{\text{for all}}~\mathbf {x} \in U} . Moreover, g {\displaystyle g} is continuously differentiable and, denoting

600-506: A + h e i ) − f ( a ) h . {\displaystyle {\begin{aligned}{\frac {\partial }{\partial x_{i}}}f(\mathbf {a} )&=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i-1},a_{i}+h,a_{i+1}\,\ldots ,a_{n})\ -f(a_{1},\ldots ,a_{i},\dots ,a_{n})}{h}}\\&=\lim _{h\to 0}{\frac {f(\mathbf {a} +h\mathbf {e_{i}} )-f(\mathbf {a} )}{h}}\,.\end{aligned}}} Where e i {\displaystyle \mathbf {e_{i}} }

700-466: A , b ) ∂ f 1 ∂ y 1 ( a , b ) ⋯ ∂ f 1 ∂ y m ( a , b ) ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ∂ f m ∂ x 1 (

800-1360: A , b ) ⋯ ∂ f m ∂ x n ( a , b ) ∂ f m ∂ y 1 ( a , b ) ⋯ ∂ f m ∂ y m ( a , b ) ] = [ X Y ] {\displaystyle (Df)(\mathbf {a} ,\mathbf {b} )=\left[{\begin{array}{ccc|ccc}{\frac {\partial f_{1}}{\partial x_{1}}}(\mathbf {a} ,\mathbf {b} )&\cdots &{\frac {\partial f_{1}}{\partial x_{n}}}(\mathbf {a} ,\mathbf {b} )&{\frac {\partial f_{1}}{\partial y_{1}}}(\mathbf {a} ,\mathbf {b} )&\cdots &{\frac {\partial f_{1}}{\partial y_{m}}}(\mathbf {a} ,\mathbf {b} )\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\{\frac {\partial f_{m}}{\partial x_{1}}}(\mathbf {a} ,\mathbf {b} )&\cdots &{\frac {\partial f_{m}}{\partial x_{n}}}(\mathbf {a} ,\mathbf {b} )&{\frac {\partial f_{m}}{\partial y_{1}}}(\mathbf {a} ,\mathbf {b} )&\cdots &{\frac {\partial f_{m}}{\partial y_{m}}}(\mathbf {a} ,\mathbf {b} )\end{array}}\right]=\left[{\begin{array}{c|c}X&Y\end{array}}\right]} where X {\displaystyle X}

900-402: A 1 , … , a i − 1 , a i + h , a i + 1 … , a n )   − f ( a 1 , … , a i , … , a n ) h = lim h → 0 f (

1000-419: A ) ) . {\displaystyle \nabla f(a)=\left({\frac {\partial f}{\partial x_{1}}}(a),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a)\right).} This vector is called the gradient of f at a . If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇ f which takes the point a to the vector ∇ f ( a ) . Consequently, the gradient produces

1100-483: A , b ) = [ ∂ f ∂ x ( a , b ) ∂ f ∂ y ( a , b ) ] = [ 2 a 2 b ] {\displaystyle (Df)(a,b)={\begin{bmatrix}{\dfrac {\partial f}{\partial x}}(a,b)&{\dfrac {\partial f}{\partial y}}(a,b)\end{bmatrix}}={\begin{bmatrix}2a&2b\end{bmatrix}}} Thus, here,

SECTION 10

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1200-446: A differentiable function of one real variable is a function whose derivative exists at each point in its domain . In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp . If x 0

1300-478: A jump discontinuity , it is possible for the derivative to have an essential discontinuity . For example, the function f ( x ) = { x 2 sin ⁡ ( 1 / x )  if  x ≠ 0 0  if  x = 0 {\displaystyle f(x)\;=\;{\begin{cases}x^{2}\sin(1/x)&{\text{ if }}x\neq 0\\0&{\text{ if }}x=0\end{cases}}}

1400-468: A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative , in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry . The partial derivative of a function f ( x , y , … ) {\displaystyle f(x,y,\dots )} with respect to

1500-392: A scalar function f ( x ) = f ( x 1 , x 2 , … , x n ) {\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\ldots ,x_{n})} along a vector v = ( v 1 , … , v n ) {\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})}

1600-2472: A vector field . A common abuse of notation is to define the del operator ( ∇ ) as follows in three-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} with unit vectors i ^ , j ^ , k ^ {\displaystyle {\hat {\mathbf {i} }},{\hat {\mathbf {j} }},{\hat {\mathbf {k} }}} : ∇ = [ ∂ ∂ x ] i ^ + [ ∂ ∂ y ] j ^ + [ ∂ ∂ z ] k ^ {\displaystyle \nabla =\left[{\frac {\partial }{\partial x}}\right]{\hat {\mathbf {i} }}+\left[{\frac {\partial }{\partial y}}\right]{\hat {\mathbf {j} }}+\left[{\frac {\partial }{\partial z}}\right]{\hat {\mathbf {k} }}} Or, more generally, for n -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} with coordinates x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} and unit vectors e ^ 1 , … , e ^ n {\displaystyle {\hat {\mathbf {e} }}_{1},\ldots ,{\hat {\mathbf {e} }}_{n}} : ∇ = ∑ j = 1 n [ ∂ ∂ x j ] e ^ j = [ ∂ ∂ x 1 ] e ^ 1 + [ ∂ ∂ x 2 ] e ^ 2 + ⋯ + [ ∂ ∂ x n ] e ^ n {\displaystyle \nabla =\sum _{j=1}^{n}\left[{\frac {\partial }{\partial x_{j}}}\right]{\hat {\mathbf {e} }}_{j}=\left[{\frac {\partial }{\partial x_{1}}}\right]{\hat {\mathbf {e} }}_{1}+\left[{\frac {\partial }{\partial x_{2}}}\right]{\hat {\mathbf {e} }}_{2}+\dots +\left[{\frac {\partial }{\partial x_{n}}}\right]{\hat {\mathbf {e} }}_{n}} The directional derivative of

1700-671: A case, evaluation of the function must be expressed in an unwieldy manner as ∂ f ( x , y , z ) ∂ x ( 17 , u + v , v 2 ) {\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}(17,u+v,v^{2})} or ∂ f ( x , y , z ) ∂ x | ( x , y , z ) = ( 17 , u + v , v 2 ) {\displaystyle \left.{\frac {\partial f(x,y,z)}{\partial x}}\right|_{(x,y,z)=(17,u+v,v^{2})}} in order to use

1800-573: A formula for f ( x , y ) . Let f : R n + m → R m {\displaystyle f:\mathbb {R} ^{n+m}\to \mathbb {R} ^{m}} be a continuously differentiable function. We think of R n + m {\displaystyle \mathbb {R} ^{n+m}} as the Cartesian product R n × R m , {\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{m},} and we write

1900-665: A function g : R n → R m {\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} whose graph ( x , g ( x ) ) {\displaystyle ({\textbf {x}},g({\textbf {x}}))} is precisely the set of all ( x , y ) {\displaystyle ({\textbf {x}},{\textbf {y}})} such that f ( x , y ) = 0 {\displaystyle f({\textbf {x}},{\textbf {y}})={\textbf {0}}} . As noted above, this may not always be possible. We will therefore fix

2000-1026: A function g : U → V {\displaystyle g:U\to V} such that the graph of g {\displaystyle g} satisfies the relation f = 0 {\displaystyle f={\textbf {0}}} on U × V {\displaystyle U\times V} , and that no other points within U × V {\displaystyle U\times V} do so. In symbols, { ( x , g ( x ) ) ∣ x ∈ U } = { ( x , y ) ∈ U × V ∣ f ( x , y ) = 0 } . {\displaystyle \{(\mathbf {x} ,g(\mathbf {x} ))\mid \mathbf {x} \in U\}=\{(\mathbf {x} ,\mathbf {y} )\in U\times V\mid f(\mathbf {x} ,\mathbf {y} )=\mathbf {0} \}.} To state

2100-415: A function is necessarily infinitely differentiable, and in fact analytic . If M is a differentiable manifold , a real or complex-valued function f on M is said to be differentiable at a point p if it is differentiable with respect to some (or any) coordinate chart defined around p . If M and N are differentiable manifolds, a function f :  M  →  N is said to be differentiable at

SECTION 20

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2200-409: A function of ( x 1 ′ , … , x m ′ ) {\displaystyle (x'_{1},\ldots ,x'_{m})} if J is invertible. Demanding J is invertible is equivalent to det J ≠ 0, thus we see that we can go back from the primed to the unprimed coordinates if the determinant of the Jacobian J is non-zero. This statement is also known as

2300-502: A function of y , that is, x = h ( y ) {\displaystyle x=h(y)} ; now the graph of the function will be ( h ( y ) , y ) {\displaystyle \left(h(y),y\right)} , since where b = 0 we have a = 1 , and the conditions to locally express the function in this form are satisfied. The implicit derivative of y with respect to x , and that of x with respect to y , can be found by totally differentiating

2400-563: A function of several variables is the case of a scalar-valued function f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} on a domain in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} (e.g., on R 2 {\displaystyle \mathbb {R} ^{2}} or R 3 {\displaystyle \mathbb {R} ^{3}} ). In this case f has

2500-472: A function that is continuous everywhere but differentiable nowhere is the Weierstrass function . A function f {\textstyle f} is said to be continuously differentiable if the derivative f ′ ( x ) {\textstyle f^{\prime }(x)} exists and is itself a continuous function. Although the derivative of a differentiable function never has

2600-462: A function. The partial derivative of f at the point a = ( a 1 , … , a n ) ∈ U {\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{n})\in U} with respect to the i -th variable x i is defined as ∂ ∂ x i f ( a ) = lim h → 0 f (

2700-403: A multi-variable function, while not being complex-differentiable. For example, f ( z ) = z + z ¯ 2 {\displaystyle f(z)={\frac {z+{\overline {z}}}{2}}} is differentiable at every point, viewed as the 2-variable real function f ( x , y ) = x {\displaystyle f(x,y)=x} , but it

2800-941: A neighbourhood of the point ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} we can write y = f ( x ) {\displaystyle y=f(x)} , where f {\displaystyle f} is a real function. Proof. Since F {\displaystyle F} is differentiable we write the differential of F {\displaystyle F} through partial derivatives: d F = grad ⁡ F ⋅ d r = ∂ F ∂ x d x + ∂ F ∂ y d y . {\displaystyle \mathrm {d} F=\operatorname {grad} F\cdot \mathrm {d} \mathbf {r} ={\frac {\partial F}{\partial x}}\mathrm {d} x+{\frac {\partial F}{\partial y}}\mathrm {d} y.} Since we are restricted to movement on

2900-479: A partial derivative ∂ f / ∂ x j {\displaystyle \partial f/\partial x_{j}} with respect to each variable x j . At the point a , these partial derivatives define the vector ∇ f ( a ) = ( ∂ f ∂ x 1 ( a ) , … , ∂ f ∂ x n (

3000-480: A point ( a , b ) = ( a 1 , … , a n , b 1 , … , b m ) {\displaystyle ({\textbf {a}},{\textbf {b}})=(a_{1},\dots ,a_{n},b_{1},\dots ,b_{m})} which satisfies f ( a , b ) = 0 {\displaystyle f({\textbf {a}},{\textbf {b}})={\textbf {0}}} , and we will ask for

3100-558: A point ( a , b ) = ( a 1 , … , a n , b 1 , … , b m ) {\displaystyle ({\textbf {a}},{\textbf {b}})=(a_{1},\dots ,a_{n},b_{1},\dots ,b_{m})} with f ( a , b ) = 0 {\displaystyle f({\textbf {a}},{\textbf {b}})=\mathbf {0} } , where 0 ∈ R m {\displaystyle \mathbf {0} \in \mathbb {R} ^{m}}

Implicit function theorem - Misplaced Pages Continue

3200-411: A point of this product as ( x , y ) = ( x 1 , … , x n , y 1 , … y m ) . {\displaystyle (\mathbf {x} ,\mathbf {y} )=(x_{1},\ldots ,x_{n},y_{1},\ldots y_{m}).} Starting from the given function f {\displaystyle f} , our goal is to construct

3300-400: A point on the curve. The statement of the theorem above can be rewritten for this simple case as follows: Theorem  —  If ∂ F ∂ y | ( x 0 , y 0 ) ≠ 0 {\displaystyle \left.{\frac {\partial F}{\partial y}}\right|_{(x_{0},y_{0})}\neq 0} then in

3400-557: A set of coordinates ( x 1 , … , x m ) {\displaystyle (x_{1},\ldots ,x_{m})} . We can introduce a new coordinate system ( x 1 ′ , … , x m ′ ) {\displaystyle (x'_{1},\ldots ,x'_{m})} by supplying m functions h 1 … h m {\displaystyle h_{1}\ldots h_{m}} each being continuously differentiable. These functions allow us to calculate

3500-917: A solution to this ODE in an open interval around the point ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} for which, at every point in it, ∂ y F ≠ 0 {\displaystyle \partial _{y}F\neq 0} . Since F {\displaystyle F} is continuously differentiable and from the assumption we have | ∂ x F | < ∞ , | ∂ y F | < ∞ , ∂ y F ≠ 0. {\displaystyle |\partial _{x}F|<\infty ,|\partial _{y}F|<\infty ,\partial _{y}F\neq 0.} From this we know that ∂ x F ∂ y F {\displaystyle {\tfrac {\partial _{x}F}{\partial _{y}F}}}

3600-436: Is analytic or continuously differentiable k {\displaystyle k} times in a neighborhood of ( a , b ) {\displaystyle ({\textbf {a}},{\textbf {b}})} , then one may choose U {\displaystyle U} in order that the same holds true for g {\displaystyle g} inside U {\displaystyle U} . In

3700-410: Is invertible , then there exists an open set U ⊂ R n {\displaystyle U\subset \mathbb {R} ^{n}} containing a {\displaystyle {\textbf {a}}} such that there exists a unique function g : U → R m {\displaystyle g:U\to \mathbb {R} ^{m}} such that g (

3800-603: Is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that f is a C function. This can be used to generalize for vector valued functions, f : U → R m {\displaystyle f:U\to \mathbb {R} ^{m}} , by carefully using a componentwise argument. The partial derivative ∂ f ∂ x {\textstyle {\frac {\partial f}{\partial x}}} can be seen as another function defined on U and can again be partially differentiated. If

3900-602: Is ∂ . One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences . The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol in 1841. Like ordinary derivatives, the partial derivative is defined as a limit . Let U be an open subset of R n {\displaystyle \mathbb {R} ^{n}} and f : U → R {\displaystyle f:U\to \mathbb {R} }

4000-460: Is a Banach space isomorphism from Y onto Z , then there exist neighbourhoods U of x 0 and V of y 0 and a Fréchet differentiable function g  : U → V such that f ( x , g ( x )) = 0 and f ( x , y ) = 0 if and only if y = g ( x ), for all ( x , y ) ∈ U × V {\displaystyle (x,y)\in U\times V} . Various forms of

4100-455: Is a function of more than one variable. For instance, z = f ( x , y ) = x 2 + x y + y 2 . {\displaystyle z=f(x,y)=x^{2}+xy+y^{2}.} The graph of this function defines a surface in Euclidean space . To every point on this surface, there are an infinite number of tangent lines . Partial differentiation

Implicit function theorem - Misplaced Pages Continue

4200-428: Is an interior point in the domain of a function f , then f is said to be differentiable at x 0 if the derivative f ′ ( x 0 ) {\displaystyle f'(x_{0})} exists. In other words, the graph of f has a non-vertical tangent line at the point ( x 0 , f ( x 0 )) . f is said to be differentiable on U if it is differentiable at every point of U . f

4300-432: Is automatically differentiable at that point, when viewed as a function f : R 2 → R 2 {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} . This is because the complex-differentiability implies that However, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } can be differentiable as

4400-499: Is continuous and bounded on both ends. From here we know that − ∂ x F ∂ y F {\displaystyle -{\tfrac {\partial _{x}F}{\partial _{y}F}}} is Lipschitz continuous in both x {\displaystyle x} and y {\displaystyle y} . Therefore, by Cauchy-Lipschitz theorem , there exists unique y ( x ) {\displaystyle y(x)} that

4500-756: Is continuous at ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} and ∂ F ∂ y | ( x 0 , y 0 ) ≠ 0 {\displaystyle \left.{\tfrac {\partial F}{\partial y}}\right|_{(x_{0},y_{0})}\neq 0} ). Therefore we have a first-order ordinary differential equation : ∂ x F d x + ∂ y F d y = 0 , y ( x 0 ) = y 0 {\displaystyle \partial _{x}F\mathrm {d} x+\partial _{y}F\mathrm {d} y=0,\quad y(x_{0})=y_{0}} Now we are looking for

4600-657: Is denoted as ∂ z ∂ x . {\displaystyle {\tfrac {\partial z}{\partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: f x ′ ( x , y , … ) , ∂ f ∂ x ( x , y , … ) . {\displaystyle f'_{x}(x,y,\ldots ),{\frac {\partial f}{\partial x}}(x,y,\ldots ).} The symbol used to denote partial derivatives

4700-896: Is differentiable at 0, since f ′ ( 0 ) = lim ε → 0 ( ε 2 sin ⁡ ( 1 / ε ) − 0 ε ) = 0 {\displaystyle f'(0)=\lim _{\varepsilon \to 0}\left({\frac {\varepsilon ^{2}\sin(1/\varepsilon )-0}{\varepsilon }}\right)=0} exists. However, for x ≠ 0 , {\displaystyle x\neq 0,} differentiation rules imply f ′ ( x ) = 2 x sin ⁡ ( 1 / x ) − cos ⁡ ( 1 / x ) , {\displaystyle f'(x)=2x\sin(1/x)-\cos(1/x)\;,} which has no limit as x → 0. {\displaystyle x\to 0.} Thus, this example shows

4800-450: Is given in the section Differentiability classes ). If f is differentiable at a point x 0 , then f must also be continuous at x 0 . In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold : a continuous function need not be differentiable. For example, a function with a bend, cusp , or vertical tangent may be continuous, but fails to be differentiable at

4900-486: Is locally one-to-one, then there exist open neighbourhoods A 0 ⊂ R n {\displaystyle A_{0}\subset \mathbb {R} ^{n}} and B 0 ⊂ R m {\displaystyle B_{0}\subset \mathbb {R} ^{m}} of x 0 and y 0 , such that, for all y ∈ B 0 {\displaystyle y\in B_{0}} ,

5000-421: Is not complex-differentiable at any point because the limit lim h → 0 h + h ¯ 2 h {\textstyle \lim _{h\to 0}{\frac {h+{\bar {h}}}{2h}}} does not exist (the limit depends on the angle of approach). Any function that is complex-differentiable in a neighborhood of a point is called holomorphic at that point. Such

5100-408: Is not necessarily differentiable, but a differentiable function is necessarily continuous (at every point where it is differentiable) as being shown below (in the section Differentiability and continuity ). A function is said to be continuously differentiable if its derivative is also a continuous function; there exist functions that are differentiable but not continuously differentiable (an example

SECTION 50

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5200-801: Is of class C 2 {\displaystyle C^{2}} if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of class C k {\displaystyle C^{k}} if the first k {\displaystyle k} derivatives f ′ ( x ) , f ′ ′ ( x ) , … , f ( k ) ( x ) {\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)} all exist and are continuous. If derivatives f ( n ) {\displaystyle f^{(n)}} exist for all positive integers n , {\textstyle n,}

5300-452: Is possible if R ≠ 0 . So it remains to check the case R = 0 . It is easy to see that in case R = 0 , our coordinate transformation is not invertible: at the origin, the value of θ is not well-defined. Based on the inverse function theorem in Banach spaces , it is possible to extend the implicit function theorem to Banach space valued mappings. Let X , Y , Z be Banach spaces . Let

5400-639: Is said to be continuously differentiable if its derivative is also a continuous function over the domain of the function f {\textstyle f} . Generally speaking, f is said to be of class C k {\displaystyle C^{k}} if its first k {\displaystyle k} derivatives f ′ ( x ) , f ′ ′ ( x ) , … , f ( k ) ( x ) {\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)} exist and are continuous over

5500-451: Is said to be differentiable at a ∈ U {\displaystyle a\in U} if the derivative exists. This implies that the function is continuous at a . This function f is said to be differentiable on U if it is differentiable at every point of U . In this case, the derivative of f is thus a function from U into R . {\displaystyle \mathbb {R} .} A continuous function

5600-518: Is the function ∇ v f {\displaystyle \nabla _{\mathbf {v} }{f}} defined by the limit ∇ v f ( x ) = lim h → 0 f ( x + h v ) − f ( x ) h . {\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\to 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.} Suppose that f

5700-402: Is the unit vector of i -th variable x i . Even if all partial derivatives ∂ f / ∂ x i ( a ) {\displaystyle \partial f/\partial x_{i}(a)} exist at a given point a , the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f

5800-434: Is the act of choosing one of these lines and finding its slope . Usually, the lines of most interest are those that are parallel to the xz -plane, and those that are parallel to the yz -plane (which result from holding either y or x constant, respectively). To find the slope of the line tangent to the function at P (1, 1) and parallel to the xz -plane, we treat y as a constant. The graph and this plane are shown on

5900-581: Is the matrix of partial derivatives in the variables x i {\displaystyle x_{i}} and Y {\displaystyle Y} is the matrix of partial derivatives in the variables y j {\displaystyle y_{j}} . The implicit function theorem says that if Y {\displaystyle Y} is an invertible matrix, then there are U {\displaystyle U} , V {\displaystyle V} , and g {\displaystyle g} as desired. Writing all

6000-409: Is the solution to the given ODE with the initial conditions. Q.E.D. Let us go back to the example of the unit circle . In this case n = m = 1 and f ( x , y ) = x 2 + y 2 − 1 {\displaystyle f(x,y)=x^{2}+y^{2}-1} . The matrix of partial derivatives is just a 1 × 2 matrix, given by ( D f ) (

6100-535: Is the zero vector. If the Jacobian matrix (this is the right-hand panel of the Jacobian matrix shown in the previous section): J f , y ( a , b ) = [ ∂ f i ∂ y j ( a , b ) ] {\displaystyle J_{f,\mathbf {y} }(\mathbf {a} ,\mathbf {b} )=\left[{\frac {\partial f_{i}}{\partial y_{j}}}(\mathbf {a} ,\mathbf {b} )\right]}

SECTION 60

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6200-419: The Y in the statement of the theorem is just the number 2 b ; the linear map defined by it is invertible if and only if b ≠ 0 . By the implicit function theorem we see that we can locally write the circle in the form y = g ( x ) for all points where y ≠ 0 . For (±1, 0) we run into trouble, as noted before. The implicit function theorem may still be applied to these two points, by writing x as

6300-472: The inverse function theorem . As a simple application of the above, consider the plane, parametrised by polar coordinates ( R , θ ) . We can go to a new coordinate system ( cartesian coordinates ) by defining functions x ( R , θ ) = R cos( θ ) and y ( R , θ ) = R sin( θ ) . This makes it possible given any point ( R , θ ) to find corresponding Cartesian coordinates ( x , y ) . When can we go back and convert Cartesian into polar coordinates? By

6400-464: The m × m identity matrix , and J is the m × m matrix of partial derivatives, evaluated at ( a , b ). (In the above, these blocks were denoted by X and Y. As it happens, in this particular application of the theorem, neither matrix depends on a .) The implicit function theorem now states that we can locally express ( x 1 , … , x m ) {\displaystyle (x_{1},\ldots ,x_{m})} as

6500-1644: The Jacobian matrix of f at a certain point ( a , b ) [ where a = ( x 1 ′ , … , x m ′ ) , b = ( x 1 , … , x m ) {\displaystyle a=(x'_{1},\ldots ,x'_{m}),b=(x_{1},\ldots ,x_{m})} ] is given by ( D f ) ( a , b ) = [ − 1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ − 1 | ∂ h 1 ∂ x 1 ( b ) ⋯ ∂ h 1 ∂ x m ( b ) ⋮ ⋱ ⋮ ∂ h m ∂ x 1 ( b ) ⋯ ∂ h m ∂ x m ( b ) ] = [ − I m | J ] . {\displaystyle (Df)(a,b)=\left[{\begin{matrix}-1&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &-1\end{matrix}}\left|{\begin{matrix}{\frac {\partial h_{1}}{\partial x_{1}}}(b)&\cdots &{\frac {\partial h_{1}}{\partial x_{m}}}(b)\\\vdots &\ddots &\vdots \\{\frac {\partial h_{m}}{\partial x_{1}}}(b)&\cdots &{\frac {\partial h_{m}}{\partial x_{m}}}(b)\\\end{matrix}}\right.\right]=[-I_{m}|J].} where I m denotes

6600-948: The Jacobian matrix of partial derivatives of g {\displaystyle g} in U {\displaystyle U} is given by the matrix product : [ ∂ g i ∂ x j ( x ) ] m × n = − [ J f , y ( x , g ( x ) ) ] m × m − 1 [ J f , x ( x , g ( x ) ) ] m × n {\displaystyle \left[{\frac {\partial g_{i}}{\partial x_{j}}}(\mathbf {x} )\right]_{m\times n}=-\left[J_{f,\mathbf {y} }(\mathbf {x} ,g(\mathbf {x} ))\right]_{m\times m}^{-1}\,\left[J_{f,\mathbf {x} }(\mathbf {x} ,g(\mathbf {x} ))\right]_{m\times n}} If, moreover, f {\displaystyle f}

6700-524: The Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with D i {\displaystyle D_{i}} as the partial derivative symbol with respect to the i -th variable. For instance, one would write D 1 f ( 17 , u + v , v 2 ) {\displaystyle D_{1}f(17,u+v,v^{2})} for

6800-517: The analytic case, this is called the analytic implicit function theorem . Suppose F : R 2 → R {\displaystyle F:\mathbb {R} ^{2}\to \mathbb {R} } is a continuously differentiable function defining a curve F ( r ) = F ( x , y ) = 0 {\displaystyle F(\mathbf {r} )=F(x,y)=0} . Let ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} be

6900-487: The curve F = 0 {\displaystyle F=0} and by assumption ∂ F ∂ y ≠ 0 {\displaystyle {\tfrac {\partial F}{\partial y}}\neq 0} around the point ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} (since ∂ F ∂ y {\displaystyle {\tfrac {\partial F}{\partial y}}}

7000-710: The direction of derivative is not repeated, it is called a mixed partial derivative . If all mixed second order partial derivatives are continuous at a point (or on a set), f is termed a C function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem : ∂ 2 f ∂ x i ∂ x j = ∂ 2 f ∂ x j ∂ x i . {\displaystyle {\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}={\frac {\partial ^{2}f}{\partial x_{j}\partial x_{i}}}.} For

7100-445: The domain of the function f {\textstyle f} . For a multivariable function, as shown here , the differentiability of it is something more complex than the existence of the partial derivatives of it. A function f : U → R {\displaystyle f:U\to \mathbb {R} } , defined on an open set U ⊂ R {\textstyle U\subset \mathbb {R} } ,

7200-521: The equation f ( x , y ) = 0 has a unique solution x = g ( y ) ∈ A 0 , {\displaystyle x=g(y)\in A_{0},} where g is a continuous function from B 0 into A 0 . Perelman’s collapsing theorem for 3-manifolds , the capstone of his proof of Thurston's geometrization conjecture , can be understood as an extension of the implicit function theorem. Partial derivative In mathematics ,

7300-711: The example described above, while the expression D 1 f {\displaystyle D_{1}f} represents the partial derivative function with respect to the first variable. For higher order partial derivatives, the partial derivative (function) of D i f {\displaystyle D_{i}f} with respect to the j -th variable is denoted D j ( D i f ) = D i , j f {\displaystyle D_{j}(D_{i}f)=D_{i,j}f} . That is, D j ∘ D i = D i , j {\displaystyle D_{j}\circ D_{i}=D_{i,j}} , so that

7400-569: The existence of a function that is differentiable but not continuously differentiable (i.e., the derivative is not a continuous function). Nevertheless, Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem . Similarly to how continuous functions are said to be of class C 0 , {\displaystyle C^{0},} continuously differentiable functions are sometimes said to be of class C 1 {\displaystyle C^{1}} . A function

7500-421: The existence of the partial derivatives (or even of all the directional derivatives ) does not guarantee that a function is differentiable at a point. For example, the function f : R → R defined by is not differentiable at (0, 0) , but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function is not differentiable at (0, 0) , but again all of

7600-2203: The following examples, let f be a function in x , y , and z . First-order partial derivatives: ∂ f ∂ x = f x ′ = ∂ x f . {\displaystyle {\frac {\partial f}{\partial x}}=f'_{x}=\partial _{x}f.} Second-order partial derivatives: ∂ 2 f ∂ x 2 = f x x ″ = ∂ x x f = ∂ x 2 f . {\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}=f''_{xx}=\partial _{xx}f=\partial _{x}^{2}f.} Second-order mixed derivatives : ∂ 2 f ∂ y ∂ x = ∂ ∂ y ( ∂ f ∂ x ) = ( f x ′ ) y ′ = f x y ″ = ∂ y x f = ∂ y ∂ x f . {\displaystyle {\frac {\partial ^{2}f}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right)=(f'_{x})'_{y}=f''_{xy}=\partial _{yx}f=\partial _{y}\partial _{x}f.} Higher-order partial and mixed derivatives: ∂ i + j + k f ∂ x i ∂ y j ∂ z k = f ( i , j , k ) = ∂ x i ∂ y j ∂ z k f . {\displaystyle {\frac {\partial ^{i+j+k}f}{\partial x^{i}\partial y^{j}\partial z^{k}}}=f^{(i,j,k)}=\partial _{x}^{i}\partial _{y}^{j}\partial _{z}^{k}f.} When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as statistical mechanics ,

7700-566: The function at a specific point are conflated by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like ∂ f ( x , y , z ) ∂ x {\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}} is used for the function, while ∂ f ( u , v , w ) ∂ u {\displaystyle {\frac {\partial f(u,v,w)}{\partial u}}} might be used for

7800-422: The function is smooth or equivalently, of class C ∞ . {\displaystyle C^{\infty }.} A function of several real variables f : R → R is said to be differentiable at a point x 0 if there exists a linear map J : R → R such that If a function is differentiable at x 0 , then all of the partial derivatives exist at x 0 , and

7900-659: The graph of y = g 1 ( x ) provides the upper half of the circle. Similarly, if g 2 ( x ) = − 1 − x 2 {\displaystyle g_{2}(x)=-{\sqrt {1-x^{2}}}} , then the graph of y = g 2 ( x ) gives the lower half of the circle. The purpose of the implicit function theorem is to tell us that functions like g 1 ( x ) and g 2 ( x ) almost always exist, even in situations where we cannot write down explicit formulas. It guarantees that g 1 ( x ) and g 2 ( x ) are differentiable, and it even works in situations where we do not have

8000-543: The graph of a function of one variable y = g ( x ) because for each choice of x ∈ (−1, 1) , there are two choices of y , namely ± 1 − x 2 {\displaystyle \pm {\sqrt {1-x^{2}}}} . However, it is possible to represent part of the circle as the graph of a function of one variable. If we let g 1 ( x ) = 1 − x 2 {\displaystyle g_{1}(x)={\sqrt {1-x^{2}}}} for −1 ≤ x ≤ 1 , then

8100-492: The hypotheses together gives the following statement. Let f : R n + m → R m {\displaystyle f:\mathbb {R} ^{n+m}\to \mathbb {R} ^{m}} be a continuously differentiable function , and let R n + m {\displaystyle \mathbb {R} ^{n+m}} have coordinates ( x , y ) {\displaystyle ({\textbf {x}},{\textbf {y}})} . Fix

8200-634: The implicit function x 2 + y 2 − 1 {\displaystyle x^{2}+y^{2}-1} and equating to 0: 2 x d x + 2 y d y = 0 , {\displaystyle 2x\,dx+2y\,dy=0,} giving d y d x = − x y {\displaystyle {\frac {dy}{dx}}=-{\frac {x}{y}}} and d x d y = − y x . {\displaystyle {\frac {dx}{dy}}=-{\frac {y}{x}}.} Suppose we have an m -dimensional space, parametrised by

8300-1097: The implicit function theorem exist for the case when the function f is not differentiable. It is standard that local strict monotonicity suffices in one dimension. The following more general form was proven by Kumagai based on an observation by Jittorntrum. Consider a continuous function f : R n × R m → R n {\displaystyle f:\mathbb {R} ^{n}\times \mathbb {R} ^{m}\to \mathbb {R} ^{n}} such that f ( x 0 , y 0 ) = 0 {\displaystyle f(x_{0},y_{0})=0} . If there exist open neighbourhoods A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} and B ⊂ R m {\displaystyle B\subset \mathbb {R} ^{m}} of x 0 and y 0 , respectively, such that, for all y in B , f ( ⋅ , y ) : A → R n {\displaystyle f(\cdot ,y):A\to \mathbb {R} ^{n}}

8400-450: The implicit function theorem, we need the Jacobian matrix of f {\displaystyle f} , which is the matrix of the partial derivatives of f {\displaystyle f} . Abbreviating ( a 1 , … , a n , b 1 , … , b m ) {\displaystyle (a_{1},\dots ,a_{n},b_{1},\dots ,b_{m})} to (

8500-413: The implicit function theorem. Ulisse Dini (1845–1918) generalized the real-variable version of the implicit function theorem to the context of functions of any number of real variables. If we define the function f ( x , y ) = x + y , then the equation f ( x , y ) = 1 cuts out the unit circle as the level set {( x , y ) | f ( x , y ) = 1} . There is no way to represent the unit circle as

8600-438: The left-hand panel of the Jacobian matrix shown in the previous section as: J f , x ( a , b ) = [ ∂ f i ∂ x j ( a , b ) ] , {\displaystyle J_{f,\mathbf {x} }(\mathbf {a} ,\mathbf {b} )=\left[{\frac {\partial f_{i}}{\partial x_{j}}}(\mathbf {a} ,\mathbf {b} )\right],}

8700-444: The linear map J is given by the Jacobian matrix , an n × m matrix in this case. A similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus. If all the partial derivatives of a function exist in a neighborhood of a point x 0 and are continuous at the point x 0 , then the function is differentiable at that point x 0 . However,

8800-421: The location of the anomaly. Most functions that occur in practice have derivatives at all points or at almost every point. However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a meagre set in the space of all continuous functions. Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of

8900-583: The mapping f  : X × Y → Z be continuously Fréchet differentiable . If ( x 0 , y 0 ) ∈ X × Y {\displaystyle (x_{0},y_{0})\in X\times Y} , f ( x 0 , y 0 ) = 0 {\displaystyle f(x_{0},y_{0})=0} , and y ↦ D f ( x 0 , y 0 ) ( 0 , y ) {\displaystyle y\mapsto Df(x_{0},y_{0})(0,y)}

9000-768: The new coordinates ( x 1 ′ , … , x m ′ ) {\displaystyle (x'_{1},\ldots ,x'_{m})} of a point, given the point's old coordinates ( x 1 , … , x m ) {\displaystyle (x_{1},\ldots ,x_{m})} using x 1 ′ = h 1 ( x 1 , … , x m ) , … , x m ′ = h m ( x 1 , … , x m ) {\displaystyle x'_{1}=h_{1}(x_{1},\ldots ,x_{m}),\ldots ,x'_{m}=h_{m}(x_{1},\ldots ,x_{m})} . One might want to verify if

9100-1401: The opposite is possible: given coordinates ( x 1 ′ , … , x m ′ ) {\displaystyle (x'_{1},\ldots ,x'_{m})} , can we 'go back' and calculate the same point's original coordinates ( x 1 , … , x m ) {\displaystyle (x_{1},\ldots ,x_{m})} ? The implicit function theorem will provide an answer to this question. The (new and old) coordinates ( x 1 ′ , … , x m ′ , x 1 , … , x m ) {\displaystyle (x'_{1},\ldots ,x'_{m},x_{1},\ldots ,x_{m})} are related by f = 0, with f ( x 1 ′ , … , x m ′ , x 1 , … , x m ) = ( h 1 ( x 1 , … , x m ) − x 1 ′ , … , h m ( x 1 , … , x m ) − x m ′ ) . {\displaystyle f(x'_{1},\ldots ,x'_{m},x_{1},\ldots ,x_{m})=(h_{1}(x_{1},\ldots ,x_{m})-x'_{1},\ldots ,h_{m}(x_{1},\ldots ,x_{m})-x'_{m}).} Now

9200-401: The partial derivative of f with respect to x , holding y and z constant, is often expressed as ( ∂ f ∂ x ) y , z . {\displaystyle \left({\frac {\partial f}{\partial x}}\right)_{y,z}.} Conventionally, for clarity and simplicity of notation, the partial derivative function and the value of

9300-420: The partial derivatives and directional derivatives exist. In complex analysis , complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers . So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x =

9400-404: The point. As these functions generally cannot be expressed in closed form , they are implicitly defined by the equations, and this motivated the name of the theorem. In other words, under a mild condition on the partial derivatives, the set of zeros of a system of equations is locally the graph of a function . Augustin-Louis Cauchy (1789–1857) is credited with the first rigorous form of

9500-1113: The previous example, it is sufficient to have det J ≠ 0 , with J = [ ∂ x ( R , θ ) ∂ R ∂ x ( R , θ ) ∂ θ ∂ y ( R , θ ) ∂ R ∂ y ( R , θ ) ∂ θ ] = [ cos ⁡ θ − R sin ⁡ θ sin ⁡ θ R cos ⁡ θ ] . {\displaystyle J={\begin{bmatrix}{\frac {\partial x(R,\theta )}{\partial R}}&{\frac {\partial x(R,\theta )}{\partial \theta }}\\{\frac {\partial y(R,\theta )}{\partial R}}&{\frac {\partial y(R,\theta )}{\partial \theta }}\\\end{bmatrix}}={\begin{bmatrix}\cos \theta &-R\sin \theta \\\sin \theta &R\cos \theta \end{bmatrix}}.} Since det J = R , conversion back to polar coordinates

9600-411: The right. Below, we see how the function looks on the plane y = 1 . By finding the derivative of the equation while assuming that y is a constant, we find that the slope of f at the point ( x , y ) is: ∂ z ∂ x = 2 x + y . {\displaystyle {\frac {\partial z}{\partial x}}=2x+y.} So at (1, 1) , by substitution,

9700-670: The slope is 3 . Therefore, ∂ z ∂ x = 3 {\displaystyle {\frac {\partial z}{\partial x}}=3} at the point (1, 1) . That is, the partial derivative of z with respect to x at (1, 1) is 3 , as shown in the graph. The function f can be reinterpreted as a family of functions of one variable indexed by the other variables: f ( x , y ) = f y ( x ) = x 2 + x y + y 2 . {\displaystyle f(x,y)=f_{y}(x)=x^{2}+xy+y^{2}.} Continuously differentiable In mathematics ,

9800-442: The value of the function at the point ( x , y , z ) = ( u , v , w ) {\displaystyle (x,y,z)=(u,v,w)} . However, this convention breaks down when we want to evaluate the partial derivative at a point like ( x , y , z ) = ( 17 , u + v , v 2 ) {\displaystyle (x,y,z)=(17,u+v,v^{2})} . In such

9900-466: The variable x {\displaystyle x} is variously denoted by It can be thought of as the rate of change of the function in the x {\displaystyle x} -direction. Sometimes, for z = f ( x , y , … ) {\displaystyle z=f(x,y,\ldots )} , the partial derivative of z {\displaystyle z} with respect to x {\displaystyle x}

10000-417: The variables are listed in the order in which the derivatives are taken, and thus, in reverse order of how the composition of operators is usually notated. Of course, Clairaut's theorem implies that D i , j = D j , i {\displaystyle D_{i,j}=D_{j,i}} as long as comparatively mild regularity conditions on f are satisfied. An important example of

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