VECTOR CALCULUS

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VECTOR CALCULUS
Subhalakshmi Lamba
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Some quantities that you have studied in your
earlier Physics Courses

• displacement
• velocity
• acceleration
• torque
• electric field

• volume
• mass
• density
• energy
• pressure

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Scalars
A scalar is a number which defines a magnitude.
It does not point to any direction in space.
50 kg 40º C
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What are vectors ?
A vector has a both a magnitude and a direction.
A physical quantity which has both a magnitude
and a direction is represented by a vector.
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Examples
B

A
Displacement Electric Field
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Vector Notation
Vectors are written as a symbol with an arrow
over the symbol
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Representation
A vector is represented by an arrow.
The arrow points in the direction of the vector.
The length of the arrow represents the
magnitude of the vector quantity.
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Representation (contd.)
It need not represent a length !!
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Addition of Vectors
Triangle Law
Head-to-tail
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Addition of Vectors

• Commutative Law

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Addition of Vectors (contd.)
Parallelogram Law
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Addition of Vectors (contd.)
Head-to-tail
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Addition of Vectors (contd.)
Delhi
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Addition of Vectors (contd.)
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Addition of Vectors (contd.)

• Associative Law

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Subtraction of Vectors
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Addition of Vectors (contd.)
Can be manipulated as in usual algebra
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• Vectors are
• geometrical objects
• independent of any
• coordinate system

Let us look for a convenient description !
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2D Cartesian Coordinates
Look a two dimensional vector in a 2D Cartesian
Coordinate System
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2D Cartesian Coordinates (contd.)
Y
ay
ax
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2D Cartesian Coordinates (contd.)

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2D Cartesian Coordinates (contd.)
Y
?
ay
?

X
ax
O
Direction cosines cos ?, cos ?
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More on Direction cosines
The direction cosines of a vector are not
independent. They satisfy the following
relation
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Vector Components
Resolving a vector The process of finding
the components of the vector.
Coordinate System
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Component of a vector
The component of the vector along an axis is
its projection along that axis
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a
3D Cartesian System
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3D Cartesian System (contd.)
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3D Cartesian System (contd.)
The direction cosines satisfy the following
relation
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Unit Vectors
Magnitude 1
Indicates direction only
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Addition of Vectors (contd.)
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Examples
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Examples
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Components
A vector can be resolved into components along
any two directions in a plane containing it.
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Components
In three dimensions, a vector can be resolved
into components along any three lines which are
non coplanar.
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SUMMARY

• A physical quantity which has
• both a magnitude and a direction
• is represented by a vector

• A geometrical representation

• An analytical description components

• Can be resolved into components along any three
  directions which are non planar.

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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