EE 543 Theory and Principles of Remote Sensing Topic 1 – Review of Vector Calculus

Outline Vectors and vector addition Unit vectors Ref: http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/waves/u10l1b.html Vectors and vector addition Unit vectors Base vectors and vector components Rectangular coordinates in 2-D Rectangular coordinates in 3-D A vector connecting two points Dot product Cross product Triple product Triple vector product Operators and Theorems O. Kilic EE543

Vector Definition  A scalar is a quantity like mass or temperature that only has a magnitude. On the other had, a vector is a mathematical object that has magnitude and direction, e.g. velocity, force. A line of given length and pointing along a given direction, such as an arrow, is the typical representation of a vector. O. Kilic EE543

Vector Notation Typical notation to designate a vector is a boldfaced character, a character with and arrow on it, or a character with a line under it (i.e., A, ). The magnitude of a vector is its length and is normally denoted by . O. Kilic EE543

Vector Addition Addition of two vectors is accomplished by laying the vectors head to tail in sequence to create a triangle. O. Kilic EE543

Vector Algebra Rules P and Q are vectors and a is a scalar O. Kilic EE543

Unit Vectors A unit vector is a vector of unit length. A unit vector is sometimes denoted by replacing the arrow on a vector with a "^" or just adding a "^" on a boldfaced character. and O. Kilic EE543

Unit Vectors(2) Any vector can be made into a unit vector by dividing it by its length. Any vector can be fully represented by providing its magnitude and a unit vector along its direction. O. Kilic EE543

Base Vectors The base vectors of a rectangular coordinate system are given by a set of three mutually orthogonal unit vectors denoted by and that are along the x, y, and z coordinate directions, respectively. O. Kilic EE543

Components Along Basis Vectors In a rectangular coordinate system the components of the vector are the projections of the vector along the x, y, and z directions. For example, in the figure the projections of vector A along the x, y, and z directions are given by Ax, Ay, and Az, respectively. The magnitude can be calculated by O. Kilic EE543

Direction Cosines The direction cosines can be calculated from the components of the vector and its magnitude through the relations O. Kilic EE543

Unit Vector Construction A unit vector can be constructed along a vector using the direction cosines as its components along the x, y, and z directions: O. Kilic EE543

Vector Connecting Two Points The vector connecting point A to point B is given by A unit vector along the line A-B can be obtained from O. Kilic EE543

Example 1 Addition of two vectors Add the two vectors: What is the magnitude of the resulting vector? What is its angle with respect to the x-axis? O. Kilic EE543

Solution 1 y A a x C B O. Kilic EE543

Example 2 Addition of three vectors: Add the vectors: What is the magnitude of the resulting vector? What is its angle with respect to the x-axis? O. Kilic EE543

Solution 2 O. Kilic EE543

Example 3 Magnitude and angles of a vector Find the magnitude and angles with respect of x, y and z axis of the vector: O. Kilic EE543

Solution 3 O. Kilic EE543

Dot Product The dot product is denoted by “.” between two vectors. The dot product of vectors A and B results in a scalar given by the relation Order is not important in the dot product Commutative O. Kilic EE543

Dot Product Properties The angle between a vector and itself is zero. Thus: Equals 1 when A = B O. Kilic EE543

Dot Product in Rectangular Coordinates i, j, k are orthogonal vectors i.i = j.j = k.k = 1 O. Kilic EE543

Example 4 Dot Product Find the dot product of the two vectors: What is the separation angle between A and B? O. Kilic EE543

Solution 4 =1 O. Kilic EE543

Projection O. Kilic EE543

Cross Product (Vector Product) The cross product of vectors a and b is a vector perpendicular to both a and b and has a magnitude equal to the area of the parallelogram generated from a and b. The direction of the cross product is given by the right-hand rule . The cross product is denoted by a “X" between the vectors O. Kilic EE543

Right Hand Rule O. Kilic EE543

Cross Product (2) O. Kilic EE543

Cross Product(3) Order is important in the cross product. If the order of operations changes in a cross product the direction of the resulting vector is reversed. O. Kilic EE543

Properties of Cross Product O. Kilic EE543

Cross Product in Rectangular Coordinates z y x Right Hand Rule O. Kilic EE543

Example Find the cross product of the two vectors: O. Kilic EE543

Solution O. Kilic EE543

Solution 2* * Valid only for Cartesian coordinates. O. Kilic EE543

The Triple Product The triple product has the following properties  The triple product of vectors a, b, and c is given by and is a scalar quantity The triple product has the following properties O. Kilic EE543

Triple Product in Rectangular Coordinates O. Kilic EE543

Triple Vector Product O. Kilic EE543

Vectors in Electromagnetics In em, we typically deal with vectors that are functions of position for a given direction. Therefore, vector components along x, y and z are not constant. The rate of change along a given direction is important in em. Electric and magnetic fields are related to each other through a differential operator. O. Kilic EE543

Main operators in vector calculus Divergence Gradient Curl Laplacian O. Kilic EE543

Vector Differentiation - Operator One of the most important and useful mathematical constructs is the "del operator", usually denoted by (which is called the "nabla"). This can be regarded as a vector whose components in the three principle directions of a Cartesian coordinate system are partial differentiations with respect to those three directions. O. Kilic EE543

Operator All the main operations of vector calculus, namely, the divergence, the gradient, the curl, and the Laplacian can be constructed from this single operator. The entities on which we operate may be either scalar fields or vector fields. O. Kilic EE543

The Gradient (Scalar to vector) If we simply multiply a scalar field such as p(x,y,z) by the del operator, the result is a vector field, and the components of the vector at each point are just the partial derivatives of the scalar field at that point, i.e., O. Kilic EE543

The Divergence (Scalar Product, Dot Product) (Vector to scalar) The divergence of a vector field v(x,y,z) = vx(x,y,z)i + vy(x,y,z)j + vz(x,y,z)k is a scalar O. Kilic EE543

The Curl (Vector Product, Cross Product) (Vector to vector) The curl of a vector field v(x,y,z) = vx(x,y,z)i + vy(x,y,z)j + vz(x,y,z)k is a vector: In Cartesian coordinates O. Kilic EE543

The Laplacian Operator (Scalar to scalar) This is sometimes called the "div grad" of a scalar field, and is given by For convenience we usually denote this operator by the symbol 2 O. Kilic EE543

Stoke’s Theorem The line integral of a vector along a closed path C is equal to the integral of the dot product of its curl and the normal to the surface which contains C as its contour. S ds dl C A O. Kilic EE543

Divergence Theorem The dot product of a vector and the normal to a closed surface S is equal to the volume integral of its divergence over the volume that is contained by S. O. Kilic EE543

References http://em-ntserver.unl.edu/Math/mathweb/vectors/vectors.html http://www.mathpages.com/home/kmath330/kmath330.htm O. Kilic EE543