Download Calculus 3 Final Exam Study Guide and Cheat Sheet and more Cheat Sheet Calculus in PDF only on Docsity! Final Exam Study Guide for Calculus III Vector Algebra 1. The length of a vector and the relationship to distances between points 2. Addition, subtraction, and scalar multiplication of vectors, together with the geometric interpretations of these operations 3. Basic properties of vector operations (p.774) 4. The dot product – definition and basic properties (p.779) 5. The geometric meaning of the dot product in terms of lengths and angles – in particular the formula a · b = |a||b| cos θ 6. Vector projections – geometric meaning and formulas 7. The cross product – definition and basic properties (#1-4 of Theorem 8, p. 790) 8. The geometric meaning of the cross product – in particular a × b is orthogonal to a and b, with magnitude |a × b| = |a||b| sin θ, and direction given by the right-hand rule 9. |a× b| is the area of the parallelogram spanned by a and b. 10. |a · (b× c)| is the volume of the parallelopiped spanned by a, b, c. 11. Cheap algebraic tests for geometric properties: • a and b are orthogonal ⇔ a · b = 0 • a and b are parallel ⇔ a× b = 0 • a, b, c are coplanar ⇔ a · (b× c) = 0 1 Lines and Planes 1. Intrinsic description (vectors) vs. Extrinsic description (scalar equa- tions) 2. Lines: passage between a vector equation, parametric equations, and symmetric equations 3. Planes: passage between a vector description (a point together with two direction vectors) and a scalar equation 4. Using vector algebra to solve geometric problems about lines and planes – it is essential that you think geometrically and try to save the number crunching in components for the last moment. Calculus of functions r : R→ Rn 1. You should be able to state and motivate the following definitions using only vector language. But you should also know the convenient compu- tational fact that everything can be checked or computed by working with component functions. • Limits and Continuity • Derivative of r(t) at a ∈ R • Definite integral of r(t) on [a, b] 2. Differentiation rules (p. 826) 3. FTC I and II for continuous r : [a, b]→ Rn 4. The tangent line to the curve traced out by r(t) at a point r(t0) – defini- tion, geometric interpretation, and ability to find equations describing it (vector, parametric, or symmetric) 5. Interpretation of a continuous r : R → R3 as describing the motion of a particle in space, in which case r′ =velocity and r′′ =acceleration 2
