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A review Analytical geometry and Vector calculus


									MECH103   Mechanisms and Dynamics of Machinery

          A review: Analytical geometry
          and Vector calculus

                   TA: Zhengjian, Xu (Eric)
                   Impact Lab: Rm1205(Lift 19)
How to describe the motion of an object?

 Shape and Location
 Flow of fluid
 Mathematic tools for mechanics
• Analytical geometry

 also called coordinate geometry and earlier referred to as
 Cartesian geometry , is the study of geometry using the
 principles of algebra.

• Vector calculus
(also called vector analysis) is concerned with multivariate real
analysis of vectors in a metric space with two or more dimensions
Coordinate system
A coordinate system is a plane (2D) or space (3D)
where the origin and axes are defined so that coordinates
can be measured.
          2D analytical geometry
          3D analytical geometry

The coordinates of a point are the components of a tuple
(sequence) of numbers (coordinate) used to represent the
location of the point in the plane or space.

In the space Rn by an n-tuple
    P = (r1, ..., rn)
    of real numbers r1, ..., rn.
    These numbers r1, ..., rn are called the coordinates of the
    point P.
Cartesian coordinate system
rectangular coordinate system

2D Coordinate system
Other coordinate systems
 Polar coordinate system
 Cylinderical coordinate system
 Spherical coordinate system

  (r, θ)
Basic contents of Analytic geometry
Important themes of analytical geometry
  Distance problems
  Line, Slope and intersection
  Description of geometrical shapes, curves
  Coordinate transformation
  Intergal and derivative
  Length, Area and volume
Calculation of Distance

                    Distance along a path compared with displacement
 Analytical geometry
 Line, slope and intersection

 Slope:                   θ


Intersection on y-axis;

Intersection on x-axis:
 Relationship between two lines
In a Cartesian coordinate system, two straight lines L and M may be
described by equations.

        L: y = ax + b,
        M: y = cx + d,
    a×c=-1                                        ax+b

Representation of a curve or surface

 Nonparametric equation (or implicit)

 Parametric equation (or explicit)
Analytical geometry
Conic sections

               ellipse   hyperbola
Analytical geometry
                Equation   Focus   Directrix
  3-D shapes


Analytical geometry
Transformation of coordinate systems
  Translation                                            y’
  ⎧x = x + h
                     ⎧x = x − h

  ⎨                  ⎨ '
  ⎩ y = y' + k       ⎩y = y − k                                            x’
                                                     O’ (h, k)                    x

 ⎧ x = x // cosθ − y // sin θ
 ⎨                                               y
 ⎩ y = x // sin θ + y // cosθ         y //
                                                              P       R         x //

 ⎧ x = x cosθ + y sin θ
     //                                                           S

 ⎨ //                                        T

 ⎩ y = − x sin θ + y cosθ
                                                     O        Q   U               x

 Important in CAD/CAM/CAE
Derivative and Integral
Differentiation and the derivative

        y = f (x)
              Δy dy
 slope =        =   = f ' ( x)
              Δx dx
dx, dy are the differentiations of x and y.       Δx

    f’(x) is called the derivative of f(x)


    b                b
        f ' ( x)dx = ∫ df ( x) = f (b) − f (a )
Table of derivatives and Integrals

        f(x)      d [f(x)] / dx
         xn         n xn-1
         ex            ex
        ln (x)        1/x
        sin x        cos x
        cos x        - sin x
        tan x        sec 2x
        cot x       - csc 2x
        sec x     sec x tan x
        csc x     - csc x cot x
Calculation about curves and shapes

           b                       b
     L=∫       (dx) + (dy ) = ∫
                   2           2
                                       1 + ( f ' ) 2 dx
          a                        a

               A = ∫ f ( x)dx

 Volume                    b
                 V = ∫ zdA

        f ( x) = x   2

     1                   1                  B
S = ∫ 1 + ( f ' ) dx = ∫ 1 + 4 x dx
                 2             2
    0                    0

           1               1 3 1 2
Area = ∫ (1 − x )dx = ( x − x ) 0 =
        0                  3        3
  Foundation of Vector Calculus
Vector and Scalar

Scalar has any magnitude,
Vector has its magnitude and orientation

Vector space: defined by a coordinate
              system and unit vectors.
In 3D rectangular coordinate system:
 Unit vector:     i= (1, 0, 0)
                  j= (0, 1, 0)
                  k= (0, 0, 1)

For example, an element of vector space Fn , x, is written:
                 F = ( x1 , x2 ,......xn )
Length (magnitude) of F:     F =      ( x1 + x 2 + ..... + x n )
                                         2      2            2
Operation of Vector
Addition and Substraction

X ± Y = ( x1 ± y1 , x2 ± y2 ,......xn ± yn )

α X = (α x , α x ,...... α x )
                 1       2         n

 0 = ( 0 , 0 ,......         0 )

 − X = ( − x ,− x ,...... − x )
             1       2       n
Dot product of Vectors
 dot product, also known as the scalar product, is
 a binary operation which takes two vectors and
 returns a scalar quantity.

 Phycial meaning in mechanics:
 For example, work done by a force F
 within a displacement d
Cross product of Vectors
cross product is: a binary operation on vectors in 3D
vector space. It is also known as the vector product. It
results in a pseudovector (Vector) rather than in a scalar.

a = a1i + a2j + a3k = [a1, a2, a3]
b = b1i + b2j + b3k = [b1, b2, b3]
a × b = [a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1].

                                Rule of Sarrus

Phycial meaning in Mechanics:
Moment of a force F with a force arm d
Vector calculus
Important themes of vector calculus
   Gradient: measures the rate and direction of change in a
   scalar field; the gradient of a scalar field is a vector field.

   Curl: measures a vector field's tendency to rotate about a
   point; the curl of a vector field is another vector field.

   Divergence: measures a vector field's tendency to originate
   from or converge upon a given point.

   Stoke’s theorem
   Divergnece theorem
These knowledges will be very useful in Thermal
Analysis and Fluid Mechanics
  Vector Calculus: Gradient
    The gradient of a scalar filed is a vector field which
    points in the direction of the greatest rate of change of
    the scalar field, and whose magnitude is the greatest
    rate of change
    The gradient of a scalar function f is denoted by:
    In Cartesian coordinate:

f = 2 x + 3 y − sin(z )

                                       In the above two images, the scalar field
                                       is in black and white, black representing
                                       higher values, and its corresponding
                                       gradient is represented by blue arrows.
           Engineering Examples

Temperature gradient and heat transfer   Air Pressure gradient and wind
Vector calculus: Curl
 Curl :
    shows a vector field’s rate of rotation about a point
 A vector field which has zero curl everywhere is called
 Apply curl on vector field F [Fx, Fy, Fz] :

                  =               =

 In a vector field that describes the linear velocities of
 each individual part of a rotating disk, the curl will have a
 constant value on all parts of the disk.
       Example for Curl
        V = {v x , v y , v z }
         v x = −v0 sin θ
         v y = v0 cos θ                            R
         vz = 0

       ⎡ i                ⎡        ⎤ ⎡     ⎤
               j    k⎤ ⎢       0   ⎥ ⎢ 0 ⎥
       ⎢∂     ∂     ∂⎥ ⎢
∇ ×V = ⎢               ⎥=      0   ⎥=⎢ 0 ⎥
       ⎢ ∂x   ∂y    ∂z ⎥ ⎢ ∂v y ∂v ⎥ ⎢ 2v0 ⎥
       ⎢ vx
       ⎣      vy    vz ⎥ ⎢     − x⎥ ⎢
                       ⎦ ⎢ ∂x ∂y ⎥ ⎣ R ⎥   ⎦
                          ⎣        ⎦

        Vector calculus: Divergence
          Divergence measures a vector field’s tendency to
          originate from or converge upon a given point.

          The divergence of a continuously differentiable
          vector field F = Fx i + Fy j + Fz k is defined to be
          the scalar-valued function

          Negative for a drain
          Positive for a well
          Zero for uniform flow
      Examples for Vector Divergence
                 V = {v x , v y , v z }
                  v x = −v0 cos θ
                 v y = −v0 sin θ

                  vz = 0

       ∂v x ∂v y ∂v z    2v0
∇ •V =     +    +     =−
       ∂x ∂y ∂z           R
     Important operation in Vector Calculus
Operation        Notation               Description                  Domain/Range
Gradient                        Measures the rate and           Maps scalar fields to
                                direction of change in a scalar vector fields.
   Curl                         Measures a vector field's         Maps vector fields to
                                tendency to rotate about a        vector fields.
Divergence                      Measures the magnitude of a Maps vector fields to
                                vector field's source or sink at scalar fields.
                                a given point.

Laplacian                       A composition of the divergence   Maps scalar fields to
                                and gradient operations.          scalar fields.
  Fundamental theorem of Vector calculus
Theorem               Statement                               Description
Gradient                                      The line integral through a gradient (vector)
theorem                                       field equals the difference in its scalar field
                                              at the endpoints of the curve.
Green's                                       The integral of the scalar curl of a vector
theorem                                       field over some region in the plane equals
                                              the line integral of the vector field over the
                                              curve bounding the region.
 Stokes'                                      The integral of the curl of a vector field
theorem                                       over a surface equals the line integral of
                                              the vector field over the curve bounding
                                              the surface.
Divergenc                                   The integral of the divergence of a vector
e theorem                                   field over some solid equals the integral of
                                            the flux through the surface bounding the

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