Below is a summary of the calculus on the plane and in space. It is arranged in a table so you see similarities and differences in different dimensions and objects. It is not meant to be a complete list of topics or formulas in Math 126. Most of the first column is from Math 124 and Math 125.

 xy-PLANE xyz-SPACE Describing Curves Describing Curves Describing Surfaces y as a function of x:  y=f(x) z as a function of x and y : z=f(x,y) implicitly: F(x,y)=0 As an intersection of two surfaces implicitly: F(x,y,z)=0 Parametrically or with a vector function: x=f(t), y=g(t) or r(t)= Parametrically or with a vector function x=f(t), y=g(t), z=h(t) or r(t)=   Here are two examples: A spiral and an ellipse Parametrically, using 2 parameters: Coming up in Math 324! Basic Example of Curves: Lines Basic Example of Curves: Lines Basic Example of Surfaces: Planes y=mx+b where m is the slope or ax+by=c. Any linear equation in x and y describes a line on the plane. Intersection of two planes a1x+b1y+c1z=d1 and    a2x+b2y+c2z=d2 ax+by+cz=d with normal vector n=. Any linear equation in x, y and z describes a plane in space.. Parametric/vector form x=x0+at, y=y0+bt or r(t)=  with direction vector v= Parametric/vector form x=x0+at, y=y0+bt, z=z0+ct  or r(t)= with direction vector v= Parametrically, using 2 parameters: Coming up in Math 324! Differential Calculus Differential Calculus Differential Calculus First derivatives and Tangent lines to curves For a parametric/vector curve the direction vector for the tangent line at t=t0 is r’(t0). First derivatives and Tangent lines to curves The direction vector for the tangent line at t=t0 is r’(t0) First derivatives and Tangent planes to surfaces If y=f(x) the equation of the tangent line at the point x=x0 is y=f(x0)+ f’(x0)(x-x0) If z=f(x,y) the equation of the tangent plane at the point (x,y)=(x0 ,y0 )is z=f(x0,y0)+ fx(x0,y0)(x-x0)+ fy(x0,y0) )(y-y0) Critical points and the second derivative test If f has a local maximum or minimum at x=a and f is differentiable at a then f’(a)=0.     Suppose f’’ is continuous near a and f’(a)=0. If f’’(a)>0, then f has a local minimum at a. If f’’(a)<0, then f has a local maximum at a. Critical points and second derivatives test If f has a local maximum or minimum at (x,y)=(a,b) and f is differentiable at (a,b) then fx(a,b)=0 and fy(a,b)=0.   Suppose fxx ,fxy, and fyy are continuous near (a,b) and fx(a,b)=0 and fy(a,b)=0. Let   If D>0 and fxx(a,b)>0, then f has a local minimum at (a,b). If D>0 and fxx(a,b)<0, then f has a local maximum at (a,b). If D<0 , then f has neither a local maximum nor a local minimum at (a,b). Optimization on closed and bounded domains If f(x) is continuous on [a,b], then it has an absolute maximum and an absolute minimum on [a,b], either at one of the critical points in (a,b) or at one of the endpoints x=a or x=b. (The “boundary” of [a,b] are its two endpoints.) Optimization on closed and bounded domains If f(x,y) is continuous on a closed and bounded domain D, then it has an absolute maximum and an absolute minimum on D,  at one of the critical points inside D or on the boundary of D. Integral Calculus Integral Calculus Integral Calculus The length of a curve r= is given by where The length of a curve r=)> is given by where If f(x)>0 on an interval I then the area under the curve y=f(x) is given by If f((x,y)>0 on a domain D then the volume under the surface  z=f(x,y) is given by