\\\\(
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Week 5, Long Day: Plane Integrals and Surface Integrals
In highschool you most likely saw integrals used for computating areas and maybe even volumes. These are only a few of the many interpretations and applications of integration. Today we shall integrate functions of two variables over very differently shaped regions in 2D space and even functions of three variables through surfaces in 3D space.
It is important that you learn to operate efficiently with parametric representations. When you master these you can design your own impressive surfaces for any application only limited by your imagination. If you e.g. want to design a new fancy teapot or an elegant skyscraper a la “Turning Torso” in Malmø, parametrizations are enormously useful and efficient. When it concerns a predefined given region or surface, you may often have to find yourself a suitable parametric representation in order to perform computations, e.g. for finding the area or maybe for computing the total mass via a given mass density function.
Today’s Key Concepts
Parametric representation of a plane in 2D space and of a surface in 3D space. Jacobian determinant and Jacobian matrix. Plane integral and surface integral. Area computation. Mass density function and mass computation. Centre of mass of weighted region.
Preparation and Syllabus
Today’s subjects are from eNote 24 Line and Plane Integrals, in particular Section 24.2, and from eNote 25 Surface and Volume Integrals, in particular Section 25.1.
Maple Ressources
Today’s Maple demo is 27_PlaneSurfaceIntegrals.
Activity Program $ $
- 10.00 – 12.00: $\,$ Lecture (aud. 42, b. 303A) (link to streaming)
- 12.00 – 17.00: $\,$ Group exercises in the study areas (b. 302, bottom floor)
- 13.00 – 16.00: $\,$ Your teachers are present
Project briefing
Today your teacher will brief you on the upcoming project work, including how to form groups, how to sign up, as well as which project topics you can choose between in prioritized order. Note that the deadline for your group’s sign up is Long Day in semester week 6.
Group Exercises
- Plane Integrals over Axis-Parallel Rectangles. By Hand
- Parametrization and Plane Integrals. By Hand
- Polar Coordinates. By Hand
- Parametric Surfaces. By Hand
- Cylindrical Surface. Parametrization and Integral
- Mass Distributions in the $(x,y)$ Plane
- Supplementary Exercise: Reparametrization