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