Math 241 Calc III Grillakis Week 1
Math 241 Calc III Grillakis Week 1 Math 241
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This 12 page Class Notes was uploaded by Colin Fields on Saturday September 3, 2016. The Class Notes belongs to Math 241 at University of Maryland taught by Dr. Grillakis in Fall 2016. Since its upload, it has received 95 views. For similar materials see Calculus 3 in Math at University of Maryland.
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Date Created: 09/03/16
Calc III Lec Day 1 8/29/16 Ch. 11.2 Vector : a set of ordered numbers Ex) vector a = (1,2,3) which does not equal (2,3,1) Uses 1. Locate points in space a. Vector a = (a ,1 ,2 )3means that from the origin (0,0,0) vector a terminates at point (a ,a1,a2) 3nd starts at point (0,0,0) 2. Vector Operations aka Algebra of Vectors a. Vector a = (a ,1 ,2 )3vector b = (b ,b 1b 2, 3 + b = (a- 1b ,1 +2 ,a2+b3) 3 b. Any number c times a vector a = (ca ,ca ,c1 ) 2 3 c. a + b = b + a 3. Diagonals of a Parallelogram a. For a parallelogram with adjacent sides formed by vectors a and b i. The long diagonal = a + b ii. The short diagonal = a – b 4. Unit Vector a. A vector with length 1 in the direction of vector a = i. This is dividing vector a by its length b. For 3 dimensional Cartesian systems, the unit vectors i, j, and k represent a vector of length 1 along the x, y, and z axis, respectively 5. Equations a. Vector a = (1,2,3) can be written as a = 1i + 2j + 3k b. In 2 dimensions, vector a can be described as where theta is the angle above the x -axis of the vector. Generalized Formulas 1. Length of an n dimensional vector a a. Calc III Lec Day 2 8/31/16 Ch. 11.3 Dot Product Properties 1. Symmetric, 2. Commutative, Uses 1. Angle between two vectors a. By solving for theta, the angle between two vectors can be computed i. 2. Directional Angles a. are the angles a vector makes with the x, y, and z axes, respectively b. Directional cosines are the cosines of these angles i. ii. iii. c. Vector a can be written as i. 3. Projection a. The projection of vector b onto vector a is the component of vector b that is parallel to vector a b. 4. Work a. The work done by a force F moving an object from point P to point Q i. 5. Law of Cosines a. For a triangle made by connecting the origin of a vector a to the terminus of a vector b, originating from the terminus of vector a, with vector c (triangle with sides a, b, c) i. ii. iii. iv. v. Theta = 90 degrees gives the Pythagorean Theorem Calc III Lec Day 3 9/2/16 Ch. 11.4 Cross Product Properties 1. a x a = 0 2. (ca) x (b) = c(a x b) = a x (cb) 3. a x (b + c) = a x b = a x c 4. a x b = -(b x a) 5. The cross produce is perpendicular to the two vectors in the cross Uses 1. Area of a Parallelogram a. The magnitude of the cross product of vectors a and b is equal to the area of a parallelogram with adjacent sides a and b b. It also follows that the cross produce is twice the area of a triangle with two sides vectors a and b 2. Finding torque a. Torque is defined as i. Where ii. PQ is the vector from the axis of rotation to where the force is being applied iii. The angle theta is made by placing the origin of the force vector at the origin of the PQ vector 3. Volume of a parallelepiped a. The volume of a parallelepiped with sides vectors a, b, and c is given by b. 4. Lagrange’s Identity a.