Notes for Analysis Et/Wi Third Quarter GS TU Delft 2001 - 2002 2/19/2019
Week 1. Surfaces and Tangential Lines 2/19/2019
Week 1. Surfaces and Tangential Planes 2/19/2019
Week 1. Parameter notation for Tangential Lines and Planes 2/19/2019
Week 1. Example: Tangent Lines but no Tangent Plane 2/19/2019
Week 1. Reminder: Differentiable in 1-d Theorem: tangent line exists differentiable 2/19/2019
Week 1. Differentiable in 2-d Definition 2/19/2019
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). 2/19/2019
Week 1. Differentiable and Partially Differentiable Theorem 2/19/2019
Week 1. Partially Differentiable and Differentiable Theorem (usually quite unworkable) 2/19/2019
Week 1. Chain Rule Theorem: the chain rule in 2-d 2/19/2019
Week 1. Example, Implicit function, a This approach is justified by the Implicit Function Theorem, which is skipped in the present course. 2/19/2019
Week 1. Example, Implicit function, b 2/19/2019
Week 1. Example, Implicit function, c 2/19/2019
Week 2. Tangent lines in other directions 2/19/2019
Week 2. Unit vector Definition 2/19/2019
Week 2. Directional derivative, the definition 2/19/2019
Week 2. Differentiable, three flavours Theorem 2/19/2019
Week 2. The gradient in 2 dimensions Definition Theorem And a similar definition and theorem for 3 dimensions …. 2/19/2019
Week 2. Tangent planes to level surfaces 2/19/2019
Week 2. The gradient and the steepest ascent/descent 2/19/2019
Week 2. The gradient field 2/19/2019
Week 2. Definition of maxima and minima And similar definitions for local and absolute minimum. 2/19/2019
Week 2. Maxima and minima 2/19/2019
Week 2. Pure 2nd order functions a, a maximum a minimum 2/19/2019
Week 2. Pure 2nd order functions b, many minima a saddle 2/19/2019
Week 2. Test for maximum, minimum or saddle 2/19/2019
Week 2. Second derivatives test Theorem (the second derivatives test) 2/19/2019
Week 2. Open and closed Definitions Open, closed and neither, but all 3 are bounded 2/19/2019
Week 2. Continuous function on closed bounded set Theorem 2/19/2019
Week 3. Integrals in two dimensions, rectangular domains I 2/19/2019
Week 3. Integrals in two dimensions, rectangular domains II 2/19/2019
Week 3. Integrals in two dimensions, rectangular domains III 2/19/2019
Week 3. Approximation by stepfunctions And we don’t care how it is defined on the edges. 2/19/2019
Week 3. The integral of a stepfunction 2/19/2019
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. 2/19/2019
Week 3. Defining the integral as a limit through stepfunctions II Definition of Riemann-integral 2/19/2019
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 2/19/2019
Week 3. Properties of the integral From now on we skip Riemann and will just say integrable. Just like the one-dimensional integral …. 2/19/2019
Week 3. The average of a function Definition: 2/19/2019
Week 3. From 2-d integral to iterated 1-d integral Fubini’s Theorem This result allows one to compute a 2-d integral. 2/19/2019
Week 3. Integrals over general domains 2/19/2019
Week 3. Type I and II domains 2/19/2019
Week 3. Other domains 2/19/2019
Week 3. Integrals over general domains, an example 2/19/2019
Week 3. Integrals over general domains: ‘the proof of the pudding is in the eating’. 2/19/2019
Week 4. Double integrals in polar coordinates 2/19/2019
Week 4. Double integrals in polar coordinates, heuristics Cartesian coordinates Polar coordinates 2/19/2019
Week 4. Double integrals to iterated integrals by polar coordinates Fubini’s Theorem in polar coordinates Sometimes such a domain is called a polar rectangle. 2/19/2019
Week 4. Integrating with polar coordinates, an example 2/19/2019
Week 4. The area of a domain 2/19/2019
Week 4. An example, the area, a 2/19/2019
Week 4. An example, the area, b 2/19/2019
Week 4. Physical applications: mass, first moments (impulsmoment) 2/19/2019
Week 4. Physical applications: the center of mass (zwaartepunt) 2/19/2019
Week 4. Physical applications: second moments (traagheidsmoment) 2/19/2019
Week 5. Stepfunctions in 3 dimensions 2/19/2019
Week 5. Defining the integral as a limit through stepfunctions I Definition of Riemann-integrable 2/19/2019
Week 5. Defining the integral as a limit through stepfunctions II Definition of Riemann-integral 2/19/2019
Week 5. Single, double and triple integrals 2/19/2019
Week 5. Recalling length, area and volume, a 2/19/2019
Week 5. Recalling length, area and volume, b 2/19/2019
Week 5. Physical applications: first moments 2/19/2019
Week 5. Physical applications: second moments 2/19/2019
Week 5. From triple integral to iterated integral. Fubini’s Theorem in 3 dimensions and there are 5 more different orders possible … . 2/19/2019
Week 5. Cylindrical coordinates 2/19/2019
Week 5. Cylindrical box 2/19/2019
Week 5. Cylindrical coordinates and Fubini and there are 5 more different orders possible … . 2/19/2019
Week 5. Spherical coordinates 2/19/2019
Week 5. Spherical coordinates again 2/19/2019
Week 5. Spherical box 2/19/2019
Week 5. Spherical coordinates and Fubini and there are 5 more different orders possible … . 2/19/2019
Week 6. Change of variables in single integrals Theorem: the substitution rule 2/19/2019
Week 6. Change of variables in 2d 2/19/2019
Week 6. Some planar geometry 2/19/2019
Week 6. Change of variables in 2d: the area transformation 2/19/2019
Week 6. A name in 2d, and two definitions Also known as injective: 2/19/2019
Week 6. Change of variables in double integrals 2/19/2019
Week 6. Change of variables in 3d, transformation of a solid 2/19/2019
Week 6. Jacobi in 3d 2/19/2019
Week 6. Change of variables in triple integrals 2/19/2019
Week 6. Example of 2d transformation: polar coordinates 2/19/2019
Week 6. Example of 3d transformation: spherical coordinates 2/19/2019
Week 6. Example of a transformation, a 2/19/2019
Week 6. Example of a transformation, b 2/19/2019