Final Review

Definitions: Memorize the following definitions. You may use either the definition given in class or the one in the book. Generally these will be the same, anyway. For any definition indicated with a *, be prepared to explain the definition with a diagram and a short essay. Pretend you are writing a section for a textbook on the definition.

Quiz 1) Resultant force (p. 804)

Quiz 2) Dot product (p. 807)
Quiz 2) Cross product (p. 814)
Quiz 2) Determinants of order 2 and 3 (p. 814)

Quiz 4) Parametric equations for a curve, parameter (p. 651)
Quiz 4) Vector valued function, component functions (p. 849)
Quiz 4) Derivative of a vector valued function (p. 856-857)
Quiz 4) Smooth curve (p. 858) *

Midterm) Arclength formula (p. 862) *
Midterm) Arclength parametrization (p. 863)
Midterm) Curvature of a curve (p. 864) *
Midterm) Unit tangent vector (p. 864)
Midterm) Unit normal vector (p. 867)

Quiz 5) Function of two variables (p. 887)
Quiz 5) Graph of a function of two variables (p. 890)
Quiz 5) Level curves (p. 892) *
Quiz 5) Limit of a function of two variables (p. 902) *
Quiz 5) Continuous function of two variables (p. 906)
Quiz 5) Partial derivatives of a function of two variables (p. 911) *
Quiz 5) Differentiable function of two variables (p. 926)
Quiz 5) Tangent Plane (p. 923)

Quiz 6) Directional Derivative (p. 941) *
Quiz 6) The gradient of a function (p. 944)
Quiz 6) Local maximum or minimum of a function of two variables (p. 953)
Quiz 6) Stationary point of a function of two variables (p. 954)

Quiz 7) Double integral of a function over a rectangle (p. 983)
Quiz 7) Average value of a function over a region (p. 986)
Quiz 7) Vector field (p. 1056, definitions 1 and 2)
Quiz 7) Gradient vector field (p. 1059)
Quiz 7) Conservative vector field and potential function (p. 1060)

Final) Line integral of a function along a path (p. 1062)

Theorems: Memorize the statements of the following theorems. If the statement is marked with a *, be prepared to explain what it means in an essay and using a diagram. If it is marked with a %, memorize the proof of it.

Quiz 1) Properties of vectors (p. 802)

Quiz 2) Properties of dot product (p. 807). Be able to prove distributivity (part 3)
Quiz 2) Law of cosines (Theorem 3, p. 808) %
Quiz 2) Orthogonality lemma (box 7, page 809)
Quiz 2) Geometric description of cross product (Theorems 5,6 p. 816, 2nd box page 817) *
Quiz 2) Parallel lemma (Corollary 7 p. 817)

Quiz 3) Parametric equations for a line parallel to v and passing through P (p. 823)
Quiz 3) Scalar equation of the plane with normal vector n and passing through P (p. 826)

Quiz 4) Geometric interpretation of the derivative of a vector valued function (p. 857) *
Quiz 4) Differentiation rules for vector valued functions (p. 859). Be able to prove 4 %
Quiz 4) The Fundamental Theorem of Calculus for vector valued functions (p. 860)

Midterm) Components of acceleration (p. 874) *, %

Quiz 5) Clairaut's Theorem (p. 916)
Quiz 5) Differentiability theorem (8, p. 926)
Quiz 5) Chain Rule, case 1 (p. 932)
Quiz 5) Chain Rule, case 2 (p. 933)
Quiz 5) General Chain Rule (p. 934)
Quiz 5) Implicit function theorem (6, p. 936)

Quiz 6) Directional derivative theorem (3, p. 942) %
Quiz 6) Geometric interpretation of the gradient (15, p. 946)%
Quiz 6) The gradient of f(x,y) is perpendicular to its level curves or surfaces (p. 947) %
Quiz 6) First derivative test for local extrema (2, p. 953)
Quiz 6) Second derivative test for local extrema (3 p. 954)
Quiz 6) Extreme value theorem for functions of two variables (8, p. 959)

Quiz 7) Method of Lagrange Multipliers (p. 966) % (in case of two variables)
Quiz 7) Method of Lagrange Multipliers for two constraints (p. 969)
Quiz 7) Fubini's Theorem

Final) no new theorems