Relation between gradient vector and partial derivatives: Difference between revisions

Revision as of 16:08, 12 May 2012

Statement

Version type	Statement
at a point, in multivariable notation	Suppose $f$ is a real-valued function of $n$ variables $x_{1}, x_{2}, \dots, x_{n}$ . Suppose $(a_{1}, a_{2}, \dots, a_{n})$ is a point in the domain of $f$ such that the gradient vector of $f$ at $(a_{1}, a_{2}, \dots, a_{n})$ , denoted $(\nabla f) (a_{1}, a_{2}, \dots, a_{n})$ , exists. Then, the partial derivatives of $f$ with respect to all variables exist, and the coordinates of the gradient vector are the partial derivatives. In other words: $(\nabla f) (a_{1}, a_{2}, \dots, a_{n}) = ⟨ f_{x_{1}} (a_{1}, a_{2}, \dots, a_{n}), f_{x_{2}} (a_{1}, a_{2}, \dots, a_{n}), \dots f_{x_{n}} (a_{1}, a_{2}, \dots, a_{n}) ⟩$
generic point, in multivariable notation	Suppose $f$ is a real-valued function of $n$ variables $x_{1}, x_{2}, \dots, x_{n}$ . Then, we have $(\nabla f) (x_{1}, x_{2}, \dots, x_{n}) = ⟨ f_{x_{1}} (x_{1}, x_{2}, \dots, x_{n}), f_{x_{2}} (x_{1}, x_{2}, \dots, x_{n}), \dots f_{x_{n}} (x_{1}, x_{2}, \dots, x_{n}) ⟩$ . Equality holds wherever the left side makes sense.
generic point, point-free notation	Suppose $f$ is a function of $n$ variables $x_{1}, x_{2}, \dots, x_{n}$ . Then, we have $\nabla f = ⟨ f_{x_{1}}, f_{x_{2}}, \dots f_{x_{n}} ⟩$ . Equality holds wherever the left side makes sense.

Related facts

Existence of partial derivatives not implies differentiable: It is possible that all the partial derivatives exist at a point but the gradient vector doesn't.
Relation between gradient vector and directional derivatives

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 ==Statement==
-Suppose <math>f</math> is a real-valued function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. Suppose <math>(a_1,a_2,\dots,a_n)</math> is a point in the domain of <math>f</math> such that the [[gradient vector]] of <math>f</math> at <math>(a_1,a_2,\dots,a_n)</math>, denoted <math>(\nabla f)(a_1,a_2,\dots,a_n)</math>, exists. Then, the [[partial derivative]]s of <math>f</math> with respect to all variables exist, and the coordinates of the gradient vector are the partial derivatives. In other words:
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+! Version type !! Statement
-<math>(\nabla f)(a_1,a_2,\dots,a_n) = \langle f_{x_1}(a_1,a_2,\dots,a_n), f_{x_2}(a_1,a_2,\dots,a_n), \dots f_{x_n}(a_1,a_2,\dots,a_n)\rangle </math>
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+| at a point, in multivariable notation || Suppose <math>f</math> is a real-valued function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. Suppose <math>(a_1,a_2,\dots,a_n)</math> is a point in the domain of <math>f</math> such that the [[gradient vector]] of <math>f</math> at <math>(a_1,a_2,\dots,a_n)</math>, denoted <math>(\nabla f)(a_1,a_2,\dots,a_n)</math>, exists. Then, the [[partial derivative]]s of <math>f</math> with respect to all variables exist, and the coordinates of the gradient vector are the partial derivatives. In other words:<br><math>(\nabla f)(a_1,a_2,\dots,a_n) = \langle f_{x_1}(a_1,a_2,\dots,a_n), f_{x_2}(a_1,a_2,\dots,a_n), \dots f_{x_n}(a_1,a_2,\dots,a_n)\rangle </math>
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+| generic point, in multivariable notation || Suppose <math>f</math> is a real-valued function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. Then, we have <br><math>(\nabla f)(x_1,x_2,\dots,x_n) = \langle f_{x_1}(x_1,x_2,\dots,x_n), f_{x_2}(x_1,x_2,\dots,x_n), \dots f_{x_n}(x_1,x_2,\dots,x_n)\rangle </math>.<br>Equality holds [[concept of equality conditional to existence of one side|wherever the left side makes sense]].
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+| generic point, point-free notation || Suppose <math>f</math> is a function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. Then, we have <br><math>\nabla f = \langle f_{x_1}, f_{x_2}, \dots f_{x_n} \rangle</math>. Equality holds [[concept of equality conditional to existence of one side|wherever the left side makes sense]].
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