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Notes for Analysis Et/Wi. Third Quarter GS TU Delft 2001 - 2002. Week 1. Surfaces and Tangential Lines. Week 1. Surfaces and Tangential Planes. Week 1. Parameter notation for Tangential Lines and Planes. Week 1. Example: Tangent Lines but no Tangent Plane.
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Notes for Analysis Et/Wi Third Quarter GS TU Delft 2001 - 2002
Week 1. Reminder:Differentiable in 1-d Theorem: tangent line exists differentiable
Week 1. Differentiable in 2-d Definition
Week 1. Linearization Definition (reminder) The linearization in a gives the tangent line to f in a. Definition The linearization in (a,b) gives the tangent plane to f in (a,b).
Week 1. Partially Differentiableand Differentiable Theorem (usually quite unworkable)
Week 1. Chain Rule Theorem: the chain rule in 2-d
Week 1. Example, Implicit function, a This approach is justified by the Implicit Function Theorem, which is skipped in the present course.
Week 2. Unit vector Definition
Week 2. The gradient in 2 dimensions Definition Theorem And a similar definition and theorem for 3 dimensions ….
Week 2. Definition ofmaxima and minima Definition Definition And similar definitions for local and absolute minimum.
Week 2. Pure 2nd order functions a, a maximum a minimum
Week 2. Pure 2nd order functions b, many minima a saddle
Week 2. Second derivatives test Theorem (the second derivatives test)
Week 2. Open and closed Definitions Open, closed and neither, but all 3 are bounded
Week 3. Approximation by stepfunctions And we don’t care how it is defined on the edges.
Week 3. Defining the integral as a limit through stepfunctions I Definition of Riemann-integrable In simple words: there exist stepfunctions above and below the function which have integrals that are arbitrary close.
Week 3. Defining the integral as a limit through stepfunctions II Definition of Riemann-integral
Week 3. Continuous functions, Riemann-sums and integrability Theorem: Continuous functions on a rectangle are Riemann-integrable and the Riemann-sums converge to the integral. a Riemann-sum
Week 3. Properties of the integral From now on we skip Riemann and will just say integrable. Just like the one-dimensional integral ….
Week 3. The average of a function Definition:
Week 3. From 2-d integral to iterated 1-d integral Fubini’s Theorem This result allows one to compute a 2-d integral.
Week 3. Integrals over general domains:‘the proof of the pudding is in the eating’.
Week 4. Double integrals in polar coordinates, heuristics Cartesian coordinates Polar coordinates
Week 4. Double integrals to iteratedintegrals by polar coordinates Fubini’s Theorem in polar coordinates Sometimes such a domain is called a polar rectangle.