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	=Vector Extreme Value Theorem=		=Vector Extreme Value Theorem=
-	<m>F: \\bbR^n \\right \\bbR^M</m>	+	<m>F: \\bbR^n \\right \\bbR^m</m>
-	<m>F</m> is continuous, <m>A \\subset \\bbR^N</m>	+	<m>F</m> is continuous, <m>A \\subset \\bbR^n</m>
	<m>\\right</m> F is bounded and obtains miminum and maximum values		<m>\\right</m> F is bounded and obtains miminum and maximum values
	i.e. <m> F(A)</m> is compact		i.e. <m> F(A)</m> is compact
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	And then I have these scribblings:		And then I have these scribblings:
-	<m> F(A) \\subset</m>	+
		+	<m> F(A)\\subset \\bigcup{i \\in I}{}{u_i} </m>
		+
		+	<m> A \\subset \\bigcup{i \\in I}{}{F^{-1}u_i} </m>
		+
		+	<m> A \\subset \\bigcup{k=1}{n}{F^{-1}u_i_k} \\doubleright  F(A) \\subset \\bigcup{k=1}{n}{u_i_k}</m>

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Revision as of 15:27, 10 February 2006

Vector Extreme Value Theorem

<m>F: \\bbR^n \\right \\bbR^m</m>
<m>F</m> is continuous, <m>A \\subset \\bbR^n</m>
<m>\\right</m> F is bounded and obtains miminum and maximum values
i.e. <m> F(A)</m> is compact

And apparently, there's no analogue to the Intermediate Value Theorem in Vector Calculus.

And then I have these scribblings:

<m> F(A)\\subset \\bigcup{i \\in I}{}{u_i} </m>

<m> A \\subset \\bigcup{i \\in I}{}{F^{-1}u_i} </m>

<m> A \\subset \\bigcup{k=1}{n}{F^{-1}u_i_k} \\doubleright F(A) \\subset \\bigcup{k=1}{n}{u_i_k}</m>