Multivariable Calculus Flashcards
Basics
f(x) = x^2 f is dependent variable x is independent variable f(x,y) = x^2 + y^2 f is dependent variable x and y is independent variable
Scalar functions
f(x,y) = x^2 + y^2
f is a scalar therefore f(x,y) is called scalar field
Vector functions
indicated as per one of the methods below arrow above vector bold matrix bracket i,j,k
Magnitude of vector
II vector II= sqrt of a1^2 +a2^2 +a3^2
is a scalar quantity
unit vector
magnitude of unit vector = II (vector with overhead hat) II usually is = 1
unit vector = vector/ magnitude of vector
Dot Product
the overlap in the magnitudes of two vectors
is a scalar quantity
vector a * vector b = a1.b1 +a2.b2+a3.b3
vector a * vector b = magnitude of a * magnitude of b *cos(delta)
Cross Product
is a vector quantity
vector a * vector b =( a2b3-b2a3) + (a3b1-a1b3)j +(a1b2-a2b1)k
resulting vector is orthogonal to both vector a and b
its direction can be determined through use of right hand rule
Coordinate systems
Cartesian coords(x,y) Polar (r,delta) they are related by x= rcos(delta) y = rsin(delta)
Parameterisation
Parameterisation = finding parametric equation of a curve or surface x^2 +y^2 = 1 x(t) = cos(t) y(t) = sin(t) over time interval 0<=t<=2pi
A space curve = the path traced out by an object that moves along that curve as a function of time
vector(r(t)) = (x(t),y(t),z(t))
vector r is a position vector as a function of time
Velocity vector
The velocity vector of an object moving along a space curve
vector(v(t)) = vector (dr/dt) = (dx/dt)i + (dy/dt)j +(dz/dt)k
The velocity vector points in the direction of travel
The speed of an object is the magnitude of its velocity vector.
Magnitude of velocity vector
II vector (v(t)) II = sqrt ( (dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2) ) This is a scalar quantity that repesents the rate of change of distance by the object.
Unit tangent vector
The velocity vector is tangential to the space curve i.e points in the direction of travel. We can therefore define a unit tangent vector (vector (torque))
vector(torque) = vector(v(t)) / II vector (v(t)) II
unit tangent vector is always = 1
Distance
The distance traveled by an object in a small increment of time
dL = II vector (v(t)) II dt
As speed can vary with time we need to add up all the small increments of distance traveled using a line integral
L = integral (with limits a- b) II vector(v(t)) II dt
Where L is distance traveled n a time interval a
Arc length
General form = integral (0-t) II vector (v(t) II dt
where s = arc length (length of space curve )
We can express a position vector in terms of arc length
vector (r (s)) = (x(s), y(s), z(s))
If vector (r) is parameterised by arc length the speed is always = 1
II vector v(t) II = 1
Acceleration
We can calculate the acceleration of an object by differentiation the velocity with respect to time .
vector (a(t)) = vector (dv/dt) = d^2x/dt^2 i + d^2y/dt^2 j + d^2z/dt^2 k
This can be seperated into two orthogonal components
acceleration = accel (tangent) + accel(norm)
The tangential acceleration points in the direction of travel . As this is the same direction as the unit tangent vector.
vector (a(tan)) = II vector(a(tan)) II * vector(torque)
magnitude of vector(a(tan))= vector a * vector torque
Circular motion
vector (r(t)) = (cost,sint)
This motion is circular and so we expect all acceleration to be in the normal direction.
Spiral motion
vector (r(t)) = (t^2. cost , t^2. sint)
We expect some acceleration in the tangential direction
Work done
As an object moves along a space curve
vector (r (t)) = (x(t) ,y(t), z(t))
It may move through a force vector field
vector (F (x,y,z)) = (Fx(x,y,z) , Fy(x,y,z), Fz(x,y,z))
By using parametric equations of the space curve we can find the force aciting on the object as a function of time.
vector(F(t)) = (Fx(t), Fy(t) , Fz(t))
we can use this to find the work done by the force
The work done by this force over a small increment of time is dw = vector(F)* vector (dr(t))
integral dw = integral (F) times integral (V)dt
A line integral is required to work out the total work done over time intgral a
