Math Tools Math Review

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Math Tools / Math Review
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Resultant Vector
Problem We wish to find the vector sum of
vectors A and B Pictorially, this is shown in
the figure on the right. Mathematically, we want
to break the vector into x, y, and maybe z
components and find the resultant vector
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System of equations

• Let 6x7y15
• And 4x-3y9
• Now find x and y which satisfy these equations
• I use method of minors

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System of Equations contd
The solution for x is found by creating a minor
wherein the constants in the equation are
substituted in place of the x value and the
value of the minor is found. It is then divided
by the value of the determinant.
I then back-substitute the value of x into my
initial equation and solve for y. 6(2.347)7y15
and solve for y (y.1304)
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Or I could use the minors again
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Larger Systems
Larger systems are broken down into their
resultant minors. For example 6x 7y 10z
12 -9x15y2z60 5x 12y-10z15
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Spherical Coordinates
Cartesian coordinates x, y, z Spherical
coordinates r, q, f Math Majors NOTE Theta!
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Cylindrical Coordinates
Cartesian coordinates x, y, z Spherical
coordinates r, f, z Math Majors NOTE Phi!
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Showing my age

• In the old days, I would tell you to use your
  integral tables
• Now, I say use your calculators to integrate
• IF YOU DARE!
• I only say this since I have seen some
  integrals which are easily found in the tables
  being integrated incorrectly by the calculator.

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Partial Derivatives

• So what is the difference between d and ?
• d like d/dx means the function only contains
  the variable x.
• When the function contains not just x but may be
  y and z, we use the partial differential,

Note that the variables y and z are held constant
when the differential operator acts on the
function What is the solution to?
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Introduction to Del

• We can now make a special differential operator
  called del. Del is defined as

• We treat del as a vector and thus, we can apply
  the dot and cross products to them.
• But first, lets recall the dot and cross
  product

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The dot or scalar product

• The scalar product is defined as the
  multiplication of two vectors in such a way that
  result is a vector
• q is the angle between A and B

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Cross or vector product

• The vector product is the multiplication of two
  vectors such that the result is a vector and
  furthermore, the resulting vector is
  perpendicular to the either of the two original
  vectors
• The best way to find a vector product is to set
  it up as a determinant as shown on the right

• q is the angle between A and B

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First application of del gradient

• The gradient is defined as the shortest or
  steepest path up a mountain or down into a
  valley.
• Lets go back to fxyz then
• You see that grad(f) makes a vector which
  points in a particular direction.
• Also, note that grad(f) takes a scalar function
  and makes a vector of it
• A particle which travels through a region of
  space wherein the potential energy, U(x,y,z),
  varies as a function of space has a force exerted
  on it equivalent to

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The scalar product and ?

• We can apply ? to the scalar product i.e.
• ?A where A is some vector
• ?A is called the divergence of A or div(A).
• Geometrically, we are discussing if A is
  diverging from some central point.

A is not diverging from a central point so Div(A)
is equal to zero
A is diverging from a central point so Div(A) is
equal to some value
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The vector product and ?

• We can apply ? to the vector product i.e.
• ?xA where A is some vector
• ?xA is called the curl of A or curl(A).
• Geometrically, we are discussing if A is curling
  around some central point.

A is curling around a central point so curl(A) is
equal to some value
A is not curling around a central point
so curl(A) is equal to zero.
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What about A ? and A x ??

• These two products do not describe the
  geometrical properties
• A ? is not equal to ? A due to the nature of
  the differential operator
• -(A ?)U would be equivalent to A F, where F is
  a force described by -?U
• Likewise for -(A x ?)U

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Two Special Integrals

• Integrating over a closed loop
• Integrating over a closed surface

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Integrating over a closed loop

• The loop can be circular or rectangular.

From 0 to 2p
Looping from Point A to Point D using straight
line segments
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Closed Surface Integral
The vector n-hat is normal to the surface. This
means that da must consist of the differential
distance in the phi direction multiplied by the
differential distance in the theta direction so
Theta is integrated from 0 to p and phi is
integrated from 0 to 2p Therefore if E depends
only on R, then