Title: Theory / Unimportant / CS / Algo / Graphics / Fundamentals -Shirley Marschner{ / 02 Math / 5 Curve and surfaces / Implicit
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Name: An implicit equation
Definition: A function whose main interest is its kernel
Page: 30
Chapter: 2
Section: 5.1
Example: \(f(x,y)=(x-x_c)^2+(y-y_c)^2-r^2\) with \((x_c,y_c)\) a point and \(r>0\) a distance
Example2: map from surface to level above/below sea. It defines the sea level

Name: The curve defined by \(f\)
Definition: The kernel of \(f\)
Page: 30
Chapter: 2
Section: 5.1
Variables: \(f\) an implicit equation

Name: The gradiant of \(f\) at \((x,y)\)
Definition: [$]\left(\pdv{f}{x},\pdv{f}{y}\right)[/$]
Page: 31
Chapter: 2
Section: 5.2
Variables: \(f:\mathbb R^2\to\mathbb R\)
\(x,y\in\mathbb R\)
Notation: \(\nabla f(x,y)\)
Definition2: vector perpendicular to the surface of \(f\), at \((x,y)\), indicating the direction of the \(f(x,y)>0\) region
Name2: The normal vector to the curve defined by \(f\) at \((x,y)\)
Extra: f ( x , y ) = 0 ??

Name: The derivative of \(g\) at \(x\)
Definition: \(\lim_{\Delta x\to 0}\frac{g(x+\Delta x)-g(x)}{\Delta x}\).
Page: 32
Chapter: 2
Section: 5.2
Variables: \(y=g(x)\) a function
\(x\in\mathbb R\)
Notation: \(g'(x)\)
Denoted by: [$]\dv{y}{x}[/$]

Name: The partial derivative of \(f\) at \(x\)
Definition: \(\lim_{\Delta x\to 0}\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}\)
Page: 32
Chapter: 2
Section: 5.2
Variables: \(f(x,y)\) a function
\(x\in\mathbb R\)
Notation: [$]\pdv{f}{x}[/$]