VECTOR CALCULUS

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VECTOR CALCULUS
Subhalakshmi Lamba
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Multiplication of a Vector

• By a scalar

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Properties
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Multiplication of Vectors(contd.)

• By a vector

two types
Vector Product
Scalar Product
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Scalar Product
DOT PRODUCT
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Scalar Product (contd.)
a b cos ?
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Scalar Product (contd.)
Magnitude of either one of the vectors
Component of the other along the direction of
the first
X
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Scalar Product (contd.)
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Properties
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Scalar Product (contd.)
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Scalar Product (contd.)
.
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Examples in Physics
W F . d
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Examples in Physics (contd.)
The rate at which work is done by a force is
the Power due to the force.
P F . v
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Examples in Physics (contd.)
A magnetic dipole moment in a magnetic field
has a potential energy, which depends on its
orientation with the field,
U (? ) - ? . B
?
B
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Some Applications

• To test whether two vectors
• are perpendicular.

If the dot product of two non zero vectors is
zero, the vectors are perpendicular.
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Some Applications

• Finding the angle between
• two vectors.

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Some Applications

• Finding the projection of one
• vector on another vector.

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Projection (contd.)
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Scalars and Vectors
A scalar is represented by a single number.
A vector is represented by a set of three
numbers. These numbers are the components of
the vectors.
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Vectors
These components are the projections of the
vector on the basis vectors of the coordinate
system chosen to describe the vectors.
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Vectors
The (numerical) values of the components will,
therefore, be different for different choices
of basis vectors.
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An example
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However , vectors have two properties that are
invariant under any (transformation) change of
coordinate axes.

• Magnitude
• Direction

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How do vectors transform?
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Vector transformation
In the original system of Cartesian coordinates.
In the rotated system of Cartesian coordinates.
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Vector transformation(contd.)
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Vector transformation(contd.)
Using
Using
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Vector transformation(contd.)
A relation between the components in the rotated
system and the original system
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Vector transformation(contd.)
A set of three numbers a i (i1,2,3) form the
components of a 3D vector only if the values of
these numbers in a rotated frame are given by
the following relations

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An example
?
?
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An example
?
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An example(contd.)
z
y'
y
x
x'
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An example(contd.)
Components of r satisfy the relation
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• SUMMARY

• Vectors as geometrical objects

• Vectors represented by components

• Addition of vectors

• Multiplication of vectors

• Scalar product

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REFERENCES

• Mathematical Methods for Physicists by George
  Arfken
• Vector Analysis by
• Murray R. Spiegel.
• Fundamentals of physics, by Halliday, Resnick and
  Walker