Multivariable Calculus

Question 1

Basics

Answer 1

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

Question 2

Scalar functions

Answer 2

f(x,y) = x^2 + y^2

f is a scalar therefore f(x,y) is called scalar field

Question 3

Vector functions

Answer 3

indicated as per one of the methods below
arrow above vector
bold 
matrix
bracket
i,j,k

Question 4

Magnitude of vector

Answer 4

II vector II= sqrt of a1^2 +a2^2 +a3^2

is a scalar quantity

Question 5

unit vector

Answer 5

magnitude of unit vector = II (vector with overhead hat) II usually is = 1
unit vector = vector/ magnitude of vector

Question 6

Dot Product

Answer 6

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)

Question 7

Cross Product

Answer 7

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

Question 8

Coordinate systems

Answer 8

Cartesian coords(x,y)
Polar (r,delta) 
they are related by x= rcos(delta)
y = rsin(delta)

Question 9

Parameterisation

Answer 9

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

Question 10

Velocity vector

Answer 10

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.

Question 11

Magnitude of velocity vector

Answer 11

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.

Question 12

Unit tangent vector

Answer 12

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

Question 13

Distance

Answer 13

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

Question 14

Arc length

Answer 14

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

Question 15

Acceleration

Answer 15

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

Question 16

Circular motion

Answer 16

vector (r(t)) = (cost,sint)

This motion is circular and so we expect all acceleration to be in the normal direction.

Question 17

Spiral motion

Answer 17

vector (r(t)) = (t^2. cost , t^2. sint)

We expect some acceleration in the tangential direction

Question 18

Work done

Answer 18

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