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Unformatted text preview: ELEC3002/ELEC7005 Vector Calculus Basic Concepts Basic Concepts (Glyn James, Ch 7.1) • Introduction (7.1) • Basic Concepts (7.1.1) • Transformations (7.1.3) ADR ELEC3002 W10/L2 2 some overlap with MATH2000 emphasising differentiation & integral aspects. Assignment in the area of Electromagnetics. ADR ELEC3002 W10/L2 3 Basic vectors (revision) In Electromagnetics, We are familiar (!?) with the fact that Electric field ( E ), Magnetic field ( H ), and Current density ( J ) are all vectors , while Potentials ( φ ), permittivity ( ε ) and permeability ( μ ) are all scalars . We also note that either vectors or scalars could be a function of position or some other variable e.g. time, frequency. So we could write ( 29 , , , E x y z t r The arrow means “vector”, sometimes written as bold we could have used another coordinate system instead! ADR ELEC3002 W10/L2 4 Vector point function Graphically, we use the arrow tipped line to represent the vector A A = r ˆ A Aa = r ˆ a unit vector magnitude of a unit vector is 1 ˆ or 1 a = we can always construct a unit vector ˆ A a A = r r k j i = = = = = = = = = z z y y x x e a z , e a y , e a x r r r ˆ ˆ ˆ ˆ ˆ ˆ ADR ELEC3002 W10/L2 5 We can always write any vector in terms of their base vectors: ˆ ˆ ˆ x y z A A x A y A z = + + r x,y , and z components of the vector A r The magnitude of the vector then becomes 2 2 2 x y z A A A A = + + r Keep in mind that generally each of the components of the vector may still be a function of x,y,z (or t or ϖ ) as well. ie. ( 29 ( 29 ( 29 ˆ ˆ ˆ , , , , , , x y z A A x y z x A x y z y A x y z z = + + r k j i = = = = = = = = = z z y y x x e a z , e a y , e a x r r r ˆ ˆ ˆ ˆ ˆ ˆ ADR ELEC3002 W10/L2 6 Position Vector In a cartesian system, a position vector r is a vector from the origin to a point (x,y,z). This is especially useful for coordinate references. x y z ˆ ˆ ˆ r xx yy zz = + + r ADR ELEC3002 W10/L2 7 so add addition & subtraction.. z z y y x x R z z y y x x R ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 2 1 1 1 1 + + = + + = r r ( 29 ( 29 ( 29 1 2 1 2 1 2 1 2 12 ˆ ˆ ˆ z z z y y y x x x R R R + + = = r r r ( 29 ( 29 ( 29 [ ] 2 1 2 1 2 2 1 2 2 1 2 12 z z y y x x R d + + = = r ADR ELEC3002 W10/L2 8 Vector addition ( 29 ) ( ˆ ) ( ˆ ˆ z z y y x x B A z B A y B A x C + + + + + = Subtraction is equivalent to the addition of A to negative B. ie. D = A – B = A + (B) ADR ELEC3002 W10/L2 9 Dot Product (scalar product) ( 29 AB B A B A θ cos r r r r = ⋅ • Always yields a scalar!...
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