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-Definition: - diff --git a/content/calculus/function-of-several-variables.md b/content/calculus/function-of-several-variables.md new file mode 100644 index 0000000..5fd177c --- /dev/null +++ b/content/calculus/function-of-several-variables.md @@ -0,0 +1,46 @@ ++++ +date = '2026-03-02T19:00:00-06:00' +draft = false +title = 'Function of Several Variables' +layout = 'chapter' +type = 'book' +tags = 'calculus' +chapterno = 15 ++++ + +## Max/Min Problems +------------------- + +Definition: +\[ +\text{Suppose } (a,b) \text{ is a point in region} R \text{ on which } f \text{ is defined and there is an open disk centered at } (a,b) \text{.}\\\\ +\text{If: } f(x,y) \leq f(a,b) \text{, then } f(a,b) \text{ is a local max.}\\ +\text{If: } f(x,y) \geq f(a,b) \text{, then } f(a,b) \text{ is a local min.} +\] + +Theorem: +\[ +\text{If } f \text{ has a local max or min value at } (a,b) \text{ and the partial derivatives } f_x \text{ and } f_y \text{ exist at } (a,b) \text{, then } f_{x}(a,b) = f_{y}(a,b) = 0 \text{.} +\] + +If you are at \((a,b)\) and every where you look, the value next to you is higher, you are at a local minimum position. If everywhere you look, the value is lower, you are at a local maximum position. + +Critical points are identified by finding where the partial derivatives of each variable in the multi-variate function equals zero. + +Definition: +\[ +\text{ A critical point in } f \text{ is located as an interior point } (a,b) \text{ if either:}\\\\ +f_{x}(a,b) = f_{y}(a,b) = 0\\ +f_{x} \text{ or } f{y} \text{ does not exist at } (a,b) +\] + +If you are able to find the critical points of \(f\) then you can use the _Second Partial Derivative Test_ to determine the maximum and minimum values. + +Theorem: +\[ +\text{If the second partial derivatives of } f \text{ are continuous throughout an open disk centered at } (a,b) \text{. Let } D(x,y) = f_{xx}(x,y)f_{yy}(x,y) - (f_{xy}(x,y))^{2} \text{.}\\\\ +\text{If } D(a,b) > 0 \text{ and } f_{xx}(a,b) < 0 \text{, there exists a local max at } (a,b) \text{.}\\ +\text{If } D(a,b) > 0 \text{ and } f_{xx}(a,b) < 0 \text{, there exists a local min at } (a,b) \text{.}\\ +\text{If } D(a,b) < 0 \text{, there exists a saddle point at }(a,b) \text{.}\\ +\text{If } D(a,b) = 0 \text{ the test is inconclusive.} +\] diff --git a/hugo.toml b/hugo.toml index 75550fd..73602ea 100644 --- a/hugo.toml +++ b/hugo.toml @@ -14,3 +14,5 @@ title = 'My New Hugo Site' [markup.goldmark.extensions.passthrough.delimiters] block = [['\[', '\]'], ['$$', '$$']] inline = [['\(', '\)']] +[params] + math = true diff --git a/layouts/_default/baseof.html b/layouts/_default/baseof.html index 0725b71..ab266bf 100644 --- a/layouts/_default/baseof.html +++ b/layouts/_default/baseof.html @@ -4,9 +4,12 @@ + {{ if .Param "math" }} + {{ partialCached "math.html" . }} + {{ end }} - +
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