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1、Scalar Product,Product of their magnitudes multiplied by the cosine of the angle between the Vectors,Scalar / Dot Product of Two Vectors,Orthogonal Vectors,Angular Dependence,Scalar Product,Scalar Product of a Vector with itself ? A . A = |A|A| cos 0 = A2,Scalar Product,Scalar Product of a Vector an
2、d Unit vector ? x . A =|x|A|cos = Ax Yields the component of a vector in a direction of the unit vector Where alpha is an angle between A and unit vector x,Scalar Product,Scalar Product of Rectangular Coordinate Unit vectors? x.y = y.z = z.x = ? = 0 x.x = y.y = z.z = ? = 1,Scalar Product Problem 3:,
3、A . B = ? ( hint: both vectors have components in three directions of unit vectors),Scalar Product Problem 4:,A = y3 + z2; B= x5 + y8 A . B = ?,Scalar Product Problem 5:,A = -x7 + y12 +z3; B = x4 + y2 + z16 A.B = ?,Line Integrals,Line Integrals,Line Integrals,Line Integrals,Line Integrals,Line Integ
4、rals,Line Integrals,Line Integrals,Spherical coordinates,Spherical coordinates,Spherical Coordinates,For many mathematical problems, it is far easier to use spherical coordinates instead of Cartesian ones. In essence, a vector r (we drop the underlining here) with the Cartesian coordinates (x,y,z) i
5、s expressed in spherical coordinates by giving its distance from the origin (assumed to be identical for both systems) |r|, and the two angles and between the direction of r and the x- and z-axis of the Cartesian system. This sounds more complicated than it actually is: and are nothing but the geogr
6、aphic longitude and latitude. The picture below illustrates this,Spherical coordinate system,Simulation of SCS,http:/ Integrals,Line Integrals,Line Integrals,Line Integrals,Tutorial,Evaluate:,Where C is right half of the circle : x2+y2=16,Solution We first need a parameterization of the circle. This
7、 is given by, We now need a range of ts that will give the right half of the circle. The following range of ts will do this:,Now, we need the derivatives of the parametric equations and lets compute ds:,Tutorial ,The line integral is then :,Assignment No 3,Q. No. 1: Evaluate where C is the curve shown below.,Assignment No 3: .,Q.NO 2: Evaluate,were C is the line segment from to,
