Vector Extreme Value Theorem
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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> compact subset |
| - | <m>\\right</m> | + | <m>\\right F(A)</m> is compact [i.e. F is bounded and attains its 'extreme' values] |
| - | + | ||
And apparently, there's no analogue to the [[Intermediate Value Theorem]] in Vector Calculus. | And apparently, there's no analogue to the [[Intermediate Value Theorem]] in Vector Calculus. | ||
Revision as of 19:01, 11 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> compact subset <m>\\right F(A)</m> is compact [i.e. F is bounded and attains its 'extreme' values]
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>
