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Applied Maths

Terms

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a.b
|a||b|cosQ a1b1+a2b2+...
a.b=0 implies
a=0 or b=0 or orthogonal
a.a
|a|^2
component of b in direction of a
b.^a^
axb
|a||b|sinQ ^n^
bxa
-(axb)
Vector eqn of a straight line
r=a+lc
Cartesian eqn of a straight line
(x-a)/l=(y-b)/m=(z-c)/n
If P divides AB in the ratio m:n then p=?
n/(m+n)a + m/(m+n)b
What is the distance between 2 non-parallel lines (one passes through A and is parallel to m, one passes through B parallel to n)
|(a-b).(mxn)|/|mxn|
What is the distance between 2 parallel lines with direction m, passing through A and B respectively?
|(a-b)xm|/|m|
Eqn of a plane using point a and normal vector n
(r-a).n=0
Eqn of a plane using point a and 2 coplanar vectors b and c
r=a+kb+lc k,l scalars
Intersection of 2 planes (r-a).m=0 and (r-b).n=0 with common point c
r=c+k(mxn)
Distance between a point P and a plane (containing point A, normal vector n)
|(a-p).(^n^)|
Triple scalar product
a.(bxc)= |a1 a2 a3| |b1 b2 c3| |c1 c2 c3|
Triple vector product
ax(bxc)= b(a.c)-c(b.a)
What is the gradient (grad) of V(x,y,z)?
(d)V/(d)x i + (d)V/(d)y j + (d)V/(d)z k
Directional derivative of V(x,y,z) in direction of n at (x,y,z)
(d)V/(d)n= (^n^).\\/V(x,y,z) ^n^.grad(V)
Curl F
\\/xF
Velocity of p wrt r v(pr)=
v(pq)+v(qr)
momentum
p=mv
Force
F=dp/dt
magnitude of force along spring
k(lenth-natural length) k(l-n)
resistive force (form)
-kv
Friction force (kinetic)
Ff=-m|R|^v^ m is coefficient of friction R is reaction force v is velocity
Moment of i at ri(t)
Mi(t)=ri(t)xFi(t)
Angular momentum of particle at r(t)
N=rxp
If N is the angular momentum of a particle, what is dN/dt?
M (moment)
Kinetic energy
T=(1/2)m|v|^2
Work done by a particle moving along r(t) t between a and b
W(a,b)=T(b)-T(a) W(r(a),r(b)= int(a,b)(F.v)dt
Conservative force field
F(r(t))=-\\/V(r) for some V(r) work done independant of path taken
Work done in a conservative force field
W(r(a),r(b))= V(r(a))-V(r(b))
If F=-\\/V then what is the potential (energy of the field)?
V
How do you tell if F is conservative?
If \\/xF=0
When is a system in equilibrium?
At a stationary point of the potential.
When is an equilibrium point stable?
When it is a minimum.
When is an equilibrium unstable?
When it is a maximum.
When is linear momentum conserved?
component conserved if no force acts in that direction.
What is conserved in an impact?
The total momentum of the system.
Newton\'s Law of Restitution
(va(after)-vb(after).n =-e(va(bef)-vb(bef).n |n|=1

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