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.
xyPLANE 
xyzSPACE 

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)=<f(t), g(t)>

Parametrically or
with a vector function x=f(t),
y=g(t), z=h(t) or r(t)=<f(t),
g(t), h(t)> 
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 a_{1}x+b_{1}y+c_{1}z=d_{1}
and a_{2}x+b_{2}y+c_{2}z=d_{2}

ax+by+cz=d
with normal vector n=<a, b, c>. Any linear equation in x, y
and z describes a plane in space.. 
Parametric/vector form x=x_{0}+at,
y=y_{0}+bt or r(t)=<x_{0}+at,
y_{0}+bt >
with direction vector v=<a, b> 
Parametric/vector
form x=x_{0}+at,
y=y_{0}+bt, z=z_{0}+ct or r(t)=<x_{0}+at,
y_{0}+bt, z_{0}+ct > with direction
vector v=<a, b, c> 
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=t_{0}
is r’(t_{0}). 
First
derivatives and Tangent lines to curves The direction
vector for the tangent line at t=t_{0} is r’(t_{0}) 
First
derivatives and Tangent planes to surfaces 
If y=f(x)
the equation of the tangent line at the point x=x_{0} is y=f(x_{0})+ f’(x_{0})(xx_{0}) 

If z=f(x,y) the equation of the tangent plane at the
point (x,y)=(x_{0} ,y_{0}
)is z=f(x_{0},y_{0})+ f_{x}(x_{0},y_{0})(xx_{0})+ f_{y}(x_{0},y_{0}) )(yy_{0}) 
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 f_{x}(a,b)=0 and f_{y}(a,b)=0. Suppose f_{xx}_{ ,}f_{xy}_{,}
and f_{yy} are continuous
near (a,b) and f_{x}(a,b)=0 and f_{y}(a,b)=0. Let If D>0
and f_{xx}(a,b)>0, then f has a local minimum
at (a,b). If D>0
and f_{xx}(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=<f(t),g(t)> is given
by where 
The length of a curve r=<f(t),g(t),h(t)>)>
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 