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**Scalar and Vector Fields**

A scalar field is a function that gives us a single value of some variable for every point in space. Examples: voltage, current, energy, temperature A vector is a quantity which has both a magnitude and a direction in space. Examples: velocity, momentum, acceleration and force

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**Example of a Scalar Field**

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Scalar Fields Week 01, Day 1 e.g. Temperature: Every location has associated value (number with units) 3 Class 01 3

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**Scalar Fields - Contours**

Week 01, Day 1 Colors represent surface temperature Contour lines show constant temperatures 4 Class 01 4

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**Fields are 3D T = T(x,y,z) Hard to visualize Work in 2D 5**

Week 01, Day 1 T = T(x,y,z) Hard to visualize Work in 2D 5 Class 01 5

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**Vector Fields Vector (magnitude, direction) at every point in space**

Week 01, Day 1 Vector Fields Vector (magnitude, direction) at every point in space Example: Velocity vector field - jet stream 6 Class 01 6

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**Vector Fields Explained**

Week 01, Day 1 Vector Fields Explained Class 01 7

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**Examples of Vector Fields**

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**Examples of Vector Fields**

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**Examples of Vector Fields**

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**Scalar and vector quantities**

Scalar quantity is defined as a quantity or parameter that has magnitude only. It independent of direction. Examples: time, temperature, volume, density, mass and energy. Vector quantity is defined as a quantity or parameter that has both magnitude and direction. Examples: velocity, electric fields and magnetic fields.

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**A vector is represent how the vector is oriented relative to some reference axis.**

Graphically, a vector is represented by an arrow, defining the direction, and the length of the arrow defines the vector’s magnitude as shown in figure. Vectors will be indicated by italic type with arrow on the character such as Ᾱ. Scalars normally are printed in italic type such as A.

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**A unit vector has a magnitude of unity (â=1).**

The unit vector in the direction of vector Ᾱ is determined by dividing A.

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By use of the unit vectors , ŷ, ẑ along x, y and z axis of a Cartesian system, a vector quantity can be written as:

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**The magnitude is defined by**

The unit vector is defined by

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Example 1 A vector Ᾱ is given as ŷ sketch Ᾱ and determines its magnitude and unit vector.

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Solution The magnitude of vector Ᾱ = ŷ is

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**Position and Distance Vectors**

A position vector is the vector from the origin of the coordinate system O (0, 0, 0) to the point P (x, y, z). It is shown as the vector The position vectors can be written as:

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A distance vector is defined as displacement of a vector from some initial point to a final point. The distance vector from P1 (x1, y1, z1) to P2 (x2, y2, z2) is The distance between two vectors is:

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Example

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Solution

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**Basic Laws of Vector Algebra**

Any number of vector quantities of the same type (i.e. same units) can be cmbined by basic vector operations. For instance, two vectors are given for vector operation below

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**Vector Addition and Subtraction**

Two vectors may be summed graphically by applying parallelogram rules or head-to-tail rule. Parallelogram rule draw both vectors from a common origin and complete the parallelogram however head-to-tail rule is obtained by placing vector at the end of vector Ᾱ to complete the triangle; either method is easily extended to three or more vectors.

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**Figure show addition of two vectors follow the rules, the sum of the addition is**

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**The rule for the subtraction of vectors follows easily from that for addition, may be expressed**

The sign, or direction of the second vector is reversed and this vector is then added to the first by the rule for vector addition

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Example Solution

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**Vector Multiplication Simple product**

Simple product multiply vectors by scalars. The magnitude of the vector changes, but its direction does not when the scalar is positive. It reverses direction when multiplied by a negative scalar.

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**Dot Product Dot product also knows as scalar product.**

It is defined as the product of the magnitude of Ᾱ, the magnitude of , and the cosine of the smaller angle, θAB between Ᾱ and . If both vectors have common origin, the sign of product is positive for the angle of 0⁰≤ θAB ≤90⁰. If the vector continued from tail of the vector, it produce negative product since the angle is 90⁰≤ θAB ≤180⁰.

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Commutative law : Distribution law : Associative law :

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**Consider two vector whose rectangular components are given such as**

Therefore yield sum of nine scalar terms, each involving the dot product of two unit vectors. Since the angle between two different unit vectors of the rectangular coordinate system is 90⁰ (cos θAB =0) so

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The remaining three terms is unity because unit vector dotted with itself since the included angle is zero (cos θAB =1) Finally, the expression with no angle is produced

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**Unit vector relationships**

It is frequently useful to resolve vectors into components along the axial directions in terms of the unit vectors i, j, and k.

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The Cross Product

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Right Hand Rule

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Example

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Solution

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Example

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Solution

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**Scalar and Vector Triple Product**

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Scalar triple product The magnitude of is the volume of the parallelepiped with edges parallel to A, B, and C. AB C B A

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Vector triple product The vector is perpendicular to the plane of A and B. When the further vector product with C is taken, the resulting vector must be perpendicular to and hence in the plane of A and B : where m and n are scalar constants to be determined. A B C AB Since this equation is valid for any vectors A, B, and C Let A = i, B = C = j:

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**VECTOR REPRESENTATION: UNIT VECTORS**

Rectangular Coordinate System x z y Unit Vector Representation for Rectangular Coordinate System The Unit Vectors imply : Points in the direction of increasing x Points in the direction of increasing y Points in the direction of increasing z

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Example

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Solution

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