21 259 Calculus in Three Dimensions Review Sheet Jemmin Chang jchang504 gmail com May 2014 Note This review sheet created by and for Jemmin Chang Topics and information are incomplete and accuracy is in no way assured Links to the web are not guaranteed to be working or useful Free to distribute without modi cation Please report any errors to the author via the email address above Contents 1 Derivatives and Integrals 1 1 Trigonometric Identities 1 2 Derivatives Integrals 1 3 2 Vectors and the Geometry of Space Stewart s Chapter 12 2 1 Vectors 2 1 1 Dot Product 2 1 2 Cross Product 2 1 3 Triple Product 2 2 Lines 2 2 1 Equations of Lines 2 3 Planes 2 3 1 Equations of Planes 2 3 2 Distance between a Point and a Plane 2 4 Quadric Surfaces 2 4 1 Equations of Quadric Surfaces 3 Vector Functions Stewart s Chapter 13 3 1 Vector Functions and Space Curves 3 2 Derivatives of Vector Functions 3 3 Arc Length 3 3 1 The Arc Length Function 3 3 2 Reparametrizing wrt Arc Length 3 4 Curvature 4 Multivariable Functions Stewart s Chapter 14 4 2 1 Techniques for Finding and Proving Limits 4 2 2 The Limit Does Not Exist 4 1 Level Curves and Surfaces 4 2 Limits and Continuity 4 3 Partial Derivatives 4 4 Tangent Planes 4 4 1 Finding an Equation of a Tangent Plane 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 1 4 4 2 Linear Approximations 4 5 Chain Rule 4 6 Directional Derivatives 4 7 The Gradient Vector 4 8 Local and Absolute Extrema 4 8 1 The Second Derivatives Test 4 8 2 Absolute Extrema 4 9 Lagrange Multipliers 5 Multiple Integrals Stewart s Chapter 15 5 1 Double Integrals Surface Area 5 2 Triple Integrals 5 1 1 6 Vector Calculus Stewart s Chapter 16 6 1 Vector Fields 6 2 Conservative Fields 6 2 1 The Fundamental Theorem for Line Integrals 6 2 2 Determining Conservatism 6 2 3 Finding the Potential Function 6 3 Line Integrals 6 3 1 Evaluating Line Integrals over Plane Curves 6 3 2 Evaluating Line Integrals over Space Curves 6 3 3 Line Integrals of Vector Fields 6 4 Green s Theorem 6 4 1 Vector Forms of Green s Theorem 6 7 1 Finding Equations of Parametric Surfaces 6 7 2 Surfaces of Revolution 6 7 3 Tangent Planes to Parametric Surfaces Surface Area 6 7 4 Surface Integrals of Vector Fields 6 9 Stokes Theorem 6 10 Divergence Theorem 6 5 Curl 6 6 Divergence 6 7 Parametric Surfaces 6 8 Surface Integrals 6 8 1 6 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 9 10 10 10 10 10 10 11 11 11 11 11 11 12 1 Derivatives and Integrals 1 1 Trigonometric Identities Inverses csc x 1 sin x sec x 1 cos x cot x 1 tan x Pythagorean sin2 x cos2 x 1 Quotient tan x sin x cos x 2 Double angle sin 2x 2 sin x cos x cos 2x cos2 x sin2 x All the useful identities can be derived easily from these See Full Table of Trig Identities 1 2 Derivatives You should have this Table of Derivatives memorized except hyperbolics dx f x g x f cid 48 x g x g cid 48 x f x Product rule d Quotient rule d Chain rule d Implicit di erentiation f x dx g x f cid 48 x g x g cid 48 x f x g x 2 dx f g x f cid 48 g x g cid 48 x 1 3 Integrals See Irina s Integration Practice and Solutions Andrew login required u substitution problem 1 above Trig identity substitution problem 3 Integration by parts problems 2 4 Can be applied multiple times Note the two sided technique in 4 Integration by partial fractions problems 5 6 Trig substitution rare but good to know 2 1 Vectors Unit vector in the direction of cid 126 v is cid 126 u cid 126 v cid 126 v Orthogonal means perpendicular 2 1 1 Dot Product cid 126 a cid 126 b cid 126 a cid 126 b cos Scalar projection of cid 126 b onto cid 126 a comp cid 126 a b cid 126 a cid 126 b cid 126 a Vector projection of cid 126 b onto cid 126 a proj cid 126 a b comp cid 126 a b cid 126 a cid 126 a 2 1 2 Cross Product cid 126 a cid 126 b cid 126 a cid 126 b sin cid 126 a and cid 126 b are parallel i cid 126 a cid 126 b cid 126 0 The area of the parallelogram determined by cid 126 a and cid 126 b is A cid 126 a cid 126 b 3 2 Vectors and the Geometry of Space Stewart s Chapter 12 2 1 3 Triple Product The volume of the parallelepiped determined by cid 126 a cid 126 b and cid 126 c is V cid 126 a cid 126 b cid 126 c cid 126 a cid 126 b cid 126 c 2 2 Lines Parallel lines have proportional direction vectors Skew lines neither intersect nor are parallel Check for intersection by setting the parametric equations equal and solving the system 2 2 1 Equations of Lines Vector equation cid 126 r cid 126 r0 t cid 126 v where r0 is the vector from the origin to any point on the line and v is the direction vector of the line Parametric equations x x0 at y y0 bt z z0 ct a b c are the direction numbers of the line i e cid 126 v cid 104 a b c cid 105 Symmetric equations x x0 Line segment from cid 126 r0 to cid 126 r1 cid 126 r t 1 t cid 126 r0 t cid 126 r1 0 t 1 a y y0 b z z0 c 2 3 Planes A plane is de ned by a point P x0 y0 z0 and a normal vector cid 126 n cid 104 a b c cid 105 Parallel planes have parallel proportional normal vectors 2 3 1 Equations of Planes Scalar equation a x x0 b y y0 c z z0 0 Linear equation ax by cz d 0 2 3 2 Distance between a Point and a Plane This distance is equal to the scalar projection of a vector cid 126 b from a point P0 on the plane to the given point P1 x1 y1 z1 onto the plane s normal vector cid 126 n cid 104 a b c cid 105 So D comp cid 126 n cid 126 b ax1 by1 cz1 d a2 b2 c2 2 4 Quadric Surfaces 2 4 1 Equations of Quadric Surfaces Note Everywhere we have …
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