{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ELEC3002-Vector Calculus 1

ELEC3002-Vector Calculus 1 - ELEC3002/ELEC7005 Vector...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
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)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
ADR ELEC3002 W10/L2 2 some overlap with MATH2000 emphasising differentiation & integral aspects. Assignment in the area of Electromagnetics.
Background image of page 2
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!
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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)
Background image of page 8
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! A cos( θ AB ) is the component of A along B. We say this is the projection of A on B.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}