Create Presentation
Download Presentation

Download Presentation
## EE 543 Theory and Principles of Remote Sensing

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**EE 543Theory and Principles of Remote Sensing**Topic 1 – Review of Vector Calculus**Outline**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 magnitudeanddirection, 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 Aunit 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 botha 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 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