# Multivariable Calculus Book Formulas - Notes from MAT397...

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Notes from MAT397 - Calculus III Notetaker: Grant Griffiths Semester: Summer Session I 2012 - Syracuse University
Contents 12 Vectors and The Geometry of Space 2 12.1 Three Dimensional Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 12.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 12.3 The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 12.4 The Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 12.5 Equations of Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 12.6 Cylinders and Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 13 Vector Functions 6 13.1 Vector Functions and Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 13.2 Derivatives and Integrals of Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 13.3 Arc Length and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 13.4 Motion in Space: Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 14 Partial Derivatives 8 14.1 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 14.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 14.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 14.4 Tangent Planes and Linear Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 14.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 14.6 Directional Derivatives and the Gradient Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 14.7 Maximum and Minimum Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 14.8 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 15 Multiple Integrals 12 15.1 Double integrals over Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 15.2 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 15.3 Double Integrals over General Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 15.4 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 15.5 Applications of Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 15.6 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 15.7 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 15.8 Triple Integrals in Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 15.9 Triple Integrals in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 15.10Change of Variables in Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 16 Vector Calculus 16 16.1 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16.2 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16.3 The Fundamental Theorem for line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16.4 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1
Chapter 12: Vectors and The Geometry of Space Lesson 12.1: Three Dimensional Coordinate Systems Coordinate Planes XY-Plane: Z is always 0 and x,y can be any real numbers XZ-Plane: Y is always 0 and x,z can be any real numbers YZ-Plane: X is always 0 and y,z can be any real numbers These three coordinate planes divide space into eight parts, call octants . The first octant is where x,y,z are all positive. A point P in R 3 with coordinates (x,y,z) is in the three-dimensional rectangular coordinate system. Distance Formula in Three Dimensions : The distance | P 1 P 2 | between the points P 1 and P 2 is | P 1 P 2 | = p ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 Equation of a Sphere: An equation of a sphere with center C(h,h,l) and radius r is ( x - h ) 2 + ( y - k ) 2 + ( z - l ) 2 = r 2 In particular, if the center is the origin O, then the equation for the sphere is x 2 + y 2 + z 2 = r 2 Lesson 12.2: Vectors Definition of Vector Addition: If u and v are vector positioned so the initial point of v is at the terminal pointer of u , then the sum v + u is the vector from the initial point of u to the terminal point of v . Definition of Scalar Multiplication: If c is a scalar and v is a vector, then the scalar multiple c v is the vector whose length is | c | times the length of v and whose direction is the same as v if c > 0 and opposite to v if c < 0. if c = 0 or v = 0, then c v =0. Vectors are parallel if they are scalar multiples of one another. If a vector has the same magnitude but opposite direction, we call it the negative of v Vector subtraction : u - v = u + ( - v ) Components of a Vector: a = h a 1 , a 2 , a 3 i where a 1 , a 2 , and a 3 are the components Position Vector: Vector from the origin to a point.
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