Chapter 1

Level curves

\(f(x,y) = k\)

Tangent planes and Normal Lines

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Vector equation of a plane
\(\textbf{r} \cdot \pmatrix{f_x(a, b)\\f_y(a,b)\\-1} = \pmatrix{a \\ b \\ f(a,b)} \cdot \pmatrix{f_x(a, b)\\f_y(a,b)\\-1}\)
OR
\(z = f_x(a,b) \cdot (x-a) + f_y(a,b) \cdot (y-b) + f(a,b)\)

In line form \(\textbf{r} =\pmatrix{a \\ b\\ f(a,b)} + \pmatrix{f_x(a,b) \\ f_y(a, b)\\ -1}t\)

Directional Derivative

\(D_u f(a, b) = \textbf{u} _1 f_x(a, b) + \textbf{u} _2 f_y(a, b) = \pmatrix{f_x(a, b) \\ f_y(a, b)} \cdot \pmatrix{\textbf{u}_1 \\ \textbf{u}_2}\)

Extrema

In some open subset of D containing (a, b) for all (x, y):

Extrema	Condition 1
Local Maximum	\(f_x(a , b) \geq f(x, y)\)
Local Minimum	\(f_y(a, b) \geq f(x, y)\)

If f has a local maximum / minimum at a point (a, b) of its domain, then
\(\huge f_x(a, b) = 0 \text{ and } f_y(a,b) = 0\)

\(f(x, y)\) has a critical point at \((a, b)\) if \(f_x (a, b) = 0\) and \(f_y (a, b) = 0\)

Derivative Test

\(\huge D = f_{xx}(a, b) \cdot f_{yy}(a,b) - (f_{xy}(a,b))^2\)

\(f\) has a local maximum at \((a, b)\) if \(D>0\) and \(f_{xx} (a,b) < 0\)
\(f\) has a local minimum at \((a, b)\) if \(D > 0\) and \(f_{xx} (a,b) > 0\)
\(f\) is neither a local maximum nor minimum at \(a,b\) if \(D<0\)

Test is inconclusive if \(D=0\)

Lagrange multiplier

The maximum / minimum value of \(f(x,y)\) subject to the constraint \(g(x,y) = 0\) occurs at the point \((x,y)\) that satisfies the following equation.

\(f_{x} = \lambda g_{x}\)

\(f_{x}=\lambda g_{y}\)

\(g(x,y) = 0\)

Next: Chapter 2