diff --git a/content/calculus/.function-of-several-variables.md.swp b/content/calculus/.function-of-several-variables.md.swp deleted file mode 100644 index 647a4c3..0000000 Binary files a/content/calculus/.function-of-several-variables.md.swp and /dev/null differ diff --git a/content/calculus/function-of-several-variables.md b/content/calculus/function-of-several-variables.md index 5fd177c..852489b 100644 --- a/content/calculus/function-of-several-variables.md +++ b/content/calculus/function-of-several-variables.md @@ -44,3 +44,19 @@ Theorem: \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.} \] + +## Lagrange Multipliers + +Theorem: +\[ +\text{Let } f \text{ be a differentiable function in } R^2 \text{ that contains curve } C \text{ given by } g(x,y) = 0. \text{ Assume } f \text{ has a local extrema on } C \text{ at point } P(a,b). \text{ Then } \nabla f(a,b) \text{ is orthogonal to the tangent line of } C \text{ at } P. \text{ Assuming } \nabla g(a,b) \neq 0, \text{ then there is a real number } \lambda \text{, or lagrange multiplier, such that } \nabla f(a,b) = \lambda \nabla g(a,b). +\] + +To find absolute extremas using lagrange multipliers: + +1. Find the gradient of \(f\) +2. Find the gradient of \(g\) +3. Set \(\nabla f(x,y) = \lambda \nabla g(x,y)\) +4. Solve the system for \(x\) and \(y\) of \(\nabla f(x,y) = \lambda \nabla g(x,y)\) and \(g(x,y) = 0\). +5. Evaluate the points you've found in \(f(x,y)\). The largest value is the absolute maximum. The smallest value is the absolute minimum. + diff --git a/layouts/_partials/math.html b/layouts/_partials/math.html index e359a19..52a7de3 100644 --- a/layouts/_partials/math.html +++ b/layouts/_partials/math.html @@ -7,13 +7,19 @@ inlineMath: [['\\(', '\\)']] // inline }, loader:{ - load: ['ui/safe'] + load: ['ui/safe'] }, output: { displayAlign: 'left', - displayIndent: '0.5in', + displayIndent: '0.25in', displayOverflow: 'linebreak' }, + linebreaks: { + inline: true, + width: '100%', + lineleading: '0.2', + LinebreakVisistor: '-' + } };