# 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