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Calculus 3
Final Flashcards
38
Mathematics
Undergraduate 1
02/22/2012
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Cards

Term
projection of vector u on v (projv(u))
 Definition
(u.v)/abs(v)^2
Term
angle between vectors u and v
 Definition
arccos(dot product/(abs(v)*abs(u))
Term
unit vector of v
 Definition
v/abs(v) or v/speed
Term
normal line from point and gradient
(vector equation for a line)
 Definition
+ t
Term
dotproduct u.v
 Definition
u1v1+u2v2+u3v3
Term
vectors are perpendicular/orthogonal if...
 Definition
u.v=0
Term
scalar projection of u on v
 Definition
((u.v)/abs(v)^2)*v
Term
Work=
 Definition
force*displacement
Term
the cross product (u x v)
 Definition
(abs(u)abs(v)sin(theta))n or the determinant
Term
ru x sv
 Definition
(rs)(uxv)
Term
u x(v+w)=
 Definition
(u x v)+(u x w)
Term
v x u=
 Definition
-(u x v)
Term
(v+w) x u=
 Definition
(v x u) + (w x u)
Term
0 x u=
 Definition
0
Term
u x (v x w)=
 Definition
(u.w)v - (u.v)w
Term
abs(u x v)
 Definition
area of a parallelogram
Term
area of triangle
 Definition
1/2 parallelogram
Term
torque vector
 Definition
abs(r)abs(F)sin(theta)n
Term
magnitude of torque
 Definition
abs(r)abs(F)sin(theta)
Term
triple scalar/box product
abs(u x v).w)
 Definition
abs(u x v)abs(w)abs(cos(theta))
area of parallelogram sided box
Term
parametric equation for a line
 Definition
through (x0,y0,z0) parallel to v=v1i+v2j+v3k
x= x0 +tv1
y= y0 +tv2
z=z+tv3
Term
distance from a point to a line in space
 Definition
abs(vectorPS x v)/(abs(v)
Term
component equation for a plane in space
 Definition
A(x-x0)+B(y-y0)+C(z-z0) =0
Term
line of intersection of planes
 Definition
find vector normal to the 2 normal vectors (take cross product) then use vector to get parametric equation of line
Term
distance from point to a plane
 Definition
abs(vectorPS . n/abs(n))
where n is vector normal to the plane
Term
angle between planes
 Definition
same as with vectors, but use normal vectors
Term
arc length differential ds=
 Definition
sqrt((dx)^2 +(dy)^2)
Term
rectangular to polar coordinates
 Definition
x=rcos(theta)
y=rsin(theta)
r^2 = x^2 + y^2
tan(theta)=y/x
P(r,theta)
Term
length of polar curve
 Definition
integral from alpha to beta of sqrt(r^2 + (dr/dtheta)^2) with respect to theta
Term
vector equation for projectile motion
 Definition
(vo cos(theta)t)i+(vo sin(theta))t-1/2gt^2)j
Term
velocity=
 Definition
derivative of position
Term
speed=
 Definition
magnitude of velocity
Term
acceleration
 Definition
derivative of velocity
Term
unit vector
 Definition
v/abs(v) in the direction of motion at time t
Term
unit tangent vector
 Definition
T= v/abs(v)
Term
curvature
 Definition
k=1/radius= 1/speed * (dT/dt)
Term
principal unit normal
 Definition
N= 1/k (dTds) = derivative of T/magnitude of derivative of T
Term
acceleration vector
 Definition
a= derivative of velocity *T +k*magnitude of speed^2 *N
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