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ELEC3002/ELEC7005 Lecture 5 Vector Calculus Basic Concepts Basic Concepts (Glyn James, Ch 7.1) Introduction (7.1) Basic Concepts (7.1.1) Transformations (7.1.3) Vector1/1 ITEE The lectures presented on Vector Calculus • emphasise differentiation & integral aspects. Applications focus on Electromagnetics. Vector1/2 ITEE 2 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 (), permittivity () and permeability () 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 we could have used E x, y , z , t The arrow means “vector”, another coordinate sometimes written as bold system instead! Vector1/3 ITEE 3 Vector point function Graphically, we use the arrow tipped line to represent the vector AA A Aa ˆ magnitude of A a unit vector is 1 we can always ˆ a construct a unit vector A or a 1 ˆ ˆ a unit vector x a x e x i, ˆ ˆ For a Cartesian system of coordinates we y a y e y j, ˆ ˆ use the following alternative notations for z a z ez k ˆ ˆ base vectors: Vector1/4 ITEE 4 We can write any vector in terms of their base vectors: A Ax x Ay y Az z ˆ ˆ ˆ x a x e x i, ˆ ˆ x, y, and z components y a y e y j, ˆ ˆ of the vector A ˆ a z ez k z ˆ The magnitude of the vector then becomes A Ax2 Ay Az2 2 Keep in mind that generally each of the components of the vector may still be a function of x,y,z (or t or ) as well. ie. A Ax x, y, z x Ay x, y, z y Az x, y, z z ˆ ˆ ˆ Vector1/5 ITEE 5 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. z r xx yy zz ˆ ˆ ˆ y x Vector1/6 ITEE 6 addition & subtraction of vectors R1 x1 x y1 y z1 z ˆ ˆ ˆ R2 x2 x y2 y z2 z ˆ ˆ ˆ the vector R12 is the vector from P1 to P2 and its distance (length or magnitude) is d: R12 R2 R1 d R12 x x2 x1 y y2 y1 z z2 z1 ˆ ˆ ˆ x2 x1 y2 y1 z 2 z1 2 2 2 12 Vector1/7 ITEE 7 Vector addition C x Ax Bx y ( Ay B y ) z ( Az Bz ) ˆ ˆ ˆ Subtraction is equivalent to the addition of A to negative B. ie. D = A – B = A + (-B) Vector1/8 ITEE 8 Dot Product (scalar product) Notice the similarity A B A B cos AB of the vector projection and the more general matrix algebra projection onto subspaces! AB is a smaller angle between A and B. • Always yields a scalar! • A cos(AB) is the component of A along B. We say this is the projection of A on B. • If two vectors are orthogonal their dot product is zero ie x y 0 ˆ ˆ • A·A=|A|2=A2 Vector1/9 ITEE 9 two definitions Note also that if two vectors are: A a1i a2 ˆ a3 k ˆ j ˆ B b1i b2 ˆ b3 k ˆ j ˆ Then the dot product can be written either as: A B A B cos AB A B a1b1 a2b2 a3b3 It can be easily shown that these two definitions are equivalent if the cosine of the included angle is expressed in terms of the direction cosines of the two vectors. Vector1/10 ITEE 10 Scalar/dot product Commutative law: A B B A Distributive law: A ( B C) A B A C Vector1/11 ITEE 11 Law of cosines for a triangle The law of cosines is a scalar relationship that expresses the length of a side of a triangle in terms of the lengths of the two other sides and the angle between them: C A2 B2 2 AB cos AB C=B-A C C C ( B A) ( B A) B B A A 2 A B 2 A2 B 2 2 AB cos AB Vector1/12 ITEE 12 Cross product (vector product) Note how the cross product produces the normal to the plane containing the two vectors! This will relate tangential and normal components. !!!!Important!!! Alternatively, the screwdriver rule: Rotating A to B moves a screw in the resultant direction Vector1/13 ITEE 13 Vector/Cross product The cross product is not commutative: A B B A The cross product is not associative: A ( B C) ( A B) C Vector1/14 ITEE 14 Cross product (ctd) ˆ x Move clockwise in the direction of the arrow the cross product is positive. Move in the counter-clockwise direction and the cross product is negative. ˆ z ˆ y x y z ˆ ˆ ˆ zx y ˆ ˆ ˆ x z y ˆ ˆ ˆ xˆ yˆ zˆ note that the cross product A B Ax Ay Az B A define the right handed coordinate Bx By Bz system x( Ay Bz Az B y ) y ( Az Bx Ax Bz ) z ( Ax B y Ay Bx ) ˆ ˆ ˆ Vector1/15 ITEE 15 Application In mechanics, the moment m of a force P about a point Q is defined as m P d where d is the (perpendicular) distance between Q and the line of action L of P. any vector from Q to any point L A on L P Q r we can now define then a moment vector d r sin m, its direction is d A perpendicular to both r and p so m r p sin or m rp mrp Vector1/16 ITEE 16 Another example In electromagnetics we deal with the flow of power by an EM wave. For any wave with an electric field E and a magnetic field H, we define a Poynting vector S given by S EH W/m 2 The direction of S is along the propagation direction (k), thus S represents the power per unit area (power density) carried by the wave. If the wave is incident on an aperture of area A with an outward unit surface vector n, then the total power intercepted by the aperture is nˆ P S n dA W ˆ kˆ A A scalar! S Vector1/17 ITEE 17 Product of Three Vectors The scalar triple product: A ( B C) B (C A) C ( A B) Note the cyclic permutation of the order of the three vectors A, B and C. The vector triple product: A ( B C) B( A C) C( A B) “BAC-CAB” rule. Vector1/18 ITEE 18 Coordinate systems: Cylindrical We commonly deal with the (x,y,z) –Cartesian coordinates. We sometimes have to deal with other orthogonal coordinate systems, for example Cylindrical or Spherical systems. unit vectors z Cylindrical are , , z ˆ ˆ ˆ Note that is always measured , , z in the x-y plane z ˆ ˆ ˆ perpendicular to z z and measured from the z ˆ ˆ ˆ x-axis! x y z ˆ ˆ ˆ A vector in a cylindrical coordinates is written as: A A A z Az ˆ ˆ ˆ Vector1/19 ITEE 19 Coordinate systems: Spherical Spherical z Again: r , , r ˆ ˆ ˆ r = azimuth = elevation x y A vector in a spherical coordinates is written as: A r Ar A A ˆ ˆ ˆ Vector1/20 ITEE 20 Transformation of Coordinates Occasionally we need to transform between different coordinate systems. Transformations of the position vector are quite simple. Some of the programming environment like Matlab can perform transformations between large number of systems. [THETA, PHI,R] = cart2sph(X,Y,Z) transforms Cartesian coordinates stored in corresponding elements of arrays X,Y and Z into spherical Coordinates. Note here that azimuth Theta and Elevation Phi are in radians. cart2pol Transform Cartesian coordinates to polar or cylindrical pol2cart Transform polar or cylindrical coordinates to Cartesian sph2cart Transform spherical coordinates to Cartesian Vector1/21 ITEE 21 Transformation of Coordinates Cylindrical to Cartesian Cartesian to Cylindrical x cos x2 y 2 y sin y tan 1 zz x zz Cartesian to Spherical Spherical to Cartesian R x2 y 2 z 2 x R sin cos x2 y 2 y R sin sin tan 1 z z R cos y tan1 x Vector1/22 ITEE 22 Jacobian matrix (James sect 7.1.3) The derivative of a coordinate transformation is the matrix of its partial derivatives. In the case of 3D coordinate systems this is always a “three by three” matrix. This matrix is sometimes called the Jacobian matrix. James considers the 2D Jacobian matrix that transforms derivatives between the (x,y) plane and the (s,t) plane. s s(x,y) u u s u t The chain rule x s x t x t t(x,y) u u s u t application may be regarded as a u f ( x, y ) y s y t y transformation becomes u F ( s, t ) u s t u x x x s u F ( s ( x, y ), t ( x, y )) u s t u y y y t Vector1/23 ITEE 23 Jacobian determinant (James sect 7.1.3) The determinant of this matrix is called the Jacobian determinant of the transformation, or else just the Jacobian. s t useful for implementing ( s, t ) x x s x t x J s t changes in variables in ( x, y ) sy t y surface & volume integrals y y This determinant measures how infinitesimal volumes change under the transformation. For this reason, the Jacobian determinant is the multiplicative factor needed to adjust the differential volume form when you change coordinates. Jacobian of the inverse transformation is x y ( x, y ) ( s , t ) ( x, y ) s 1 J1 s xs ys ( s , t ) ( x, y ) ( s, t ) x y xt yt t t or J1 J 1 Vector1/24 ITEE 24 Jacobian matrix and determinant Transformation from x-y s 2x y Example: plane to s-t plane t x 2y s t x x 2 1 J 5 s t 1 2 1 y y x (2s t ) 5 If we now invert the transformation 1 y ( s 2t ) x y 2 1 5 s s 5 5 1 Note that if J=J1 =0, then the J1 x y 1 2 5 variables are functionally dependent t t 5 5 and imply a non-unique correspondence between the Confirming that J1=J-1 (x,y) and (s,t) planes Vector1/25 ITEE 25 Jacobian determinant (James sect 7.1.3) The definition of a Jacobian is not restricted to functions of two variables. For example, for functions of three variables: u U ( x, y, z ); v V ( x, y, z ); w W ( x, y, z ); the Jacobian matrix and its determinant is obtained from : u v w x x x u x v x wx (u , v, w) u v w J u y v y wy ( x, y, z ) y y y u z v z wz u v w z z z Vector1/26 ITEE 26 Jacobian determinant Example: Show that if x+y=u and y=uv ( x, y ) J u (u , v) Solution: x u y u uv y uv x y ( x, y ) u u 1 v v J (1 v)u uv u (u, v) x y u u v v Vector1/27 ITEE 27 Jacobian determinant Example: x e u cos v y e u sin v ( x, y ) (u , v) Determine : J and J1 (u, v) ( x, y ) And verify that they are mutual inverses. x y Solution: ( x, y ) u u J e 2u (u, v) x y v v (u , v) J1 ( x, y ) 1 1 2 2u x y2 e J J1 1 Vector1/28 ITEE 28 Curves in space The parametric representation of a curve in space is the most useful. Basically, the point in space (x,y,z) is a function of a parameter t. x x(t ), y y (t ), z z (t ) We let t vary over some range and the points trace out a curve. To put it in a vectorial form we draw a position vector to the point on the curve C. z C Note that this gives r(t) some orientation to the curve +ve/- depending on whether the parameter t is x y increasing or decreasing. Vector1/29 ITEE 29 Some examples For example a helix: r t 6 cos t x 6sin t y t z ˆ ˆ ˆ Point moves in a circular fashion in x-y and then along the z direction: A straight line through a point A with a position vector a in the direction of a constant vector b is: z bA r t a tb a x y Vector1/30 ITEE 30 unit tangents A tangent (vector) to the curve is given by the derivative of the curve vector wrt the parameter. e.g. A tangent vector to the helix r t 6 cos t x 6sin t y t z is given by : ˆ ˆ ˆ r t 6sin t x 6 cos t y z ˆ ˆ ˆ A unit tangent vector is obtained from the tangent vector by normalization (ie dividing r'(t) by its length). r Tangents are very useful, u we use them to approximate r curves with straight lines! Vector1/31 ITEE 31 Arc Length The arc length along a curve is also a useful quantity. We define this in terms of the magnitude of the derivative: b b length variable s s (t ) r r dt r dt a a Example: for a curve r (t ) t 2 x t 3 y ˆ ˆ r ' (t ) 2t x 3t 2 y t (2 x 3ty ) ˆ ˆ ˆ ˆ r ' (t ) r ' (t ) (2 x 3ty ) (2 x 3ty ) t 4 9t 2 ˆ ˆ ˆ ˆ we can show that the length of the curve starting at the point t = 0 to some point p is: p s t t 4 9t 2 dt 0 Vector1/32 ITEE 32 Arc length as a parameter Instead of using the parameter t along the curve we can use the arc length s instead! If we do this, the tangent vector r'(s) is the unit tangent vector. Note the It is reasonably easy to parameterise a curve in terms variable ‘s’ of the arc length: 1. Get s(t) by computing the arc length integral from 0 to t. 2. Invert the function s =s(t) to t = t(s) 3. Write x = x(s) = x(t(s)), y = y(s) = y(t(s)), to obtain r(s) 4. The magnitude of the tangent will be 1. Prove this by taking the derivative. Vector1/33 ITEE 33 Tangent, normal and binormal There are other vectors associated with curves such as a normal to the curve at a point and a binormal vector that forms a 3D coordinate system along the tangent vector at any point along the curve. normal These are used mainly in dynamics and optics giving rise to concepts of z tangent velocity acceleration, curvature and torsion etc. binormal x y Vector1/34 ITEE 34