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Week 7, Short Day: More about Gradient Vector Fields

Today we refine the investigation of vector fields, in particular gradient vector fields. We shall enjoy some beautiful theorems about tangential line integrals of gradient vector fields.

  • For example the fact that for a gradient vector field the tangential line integral is independent of the path of integration.
  • And the fact that if you have an antiderivative, then you can find the tangential line integral using the interval end point values of the indefinite integral.
  • And the fact that for a gradient field any circulation will be zero.
  • And the fact that the curl of a gradient field is zero.

For these and also for building towards next week’s main topic theorems we will today introduce the concepts divergence and curl as measures of different properties of a vector field.

Today’s Key Concepts

Gradient vector field theorems. Indefinite integrals and The Fundamental Theorem of Algebra. Divergence $\mathrm{Div}(\mV)$. Curl $\mathrm{Curl}(\mV)$. Circulation $\mathrm{Circ}(\mV,\mathcal K)$.

Preparation and Syllabus

Today we will continue with subjects from eNote 26 Vector Fields and eNote 27 Vector Fields Along Curves about the tangential line intgral.

Maple Syllabus

Today’s Maple demo is 31_DivCurl.

Activity program

  • 13:00 – 14.00: $\,$ Lecture (aud. 42, b. 303A) (link to streaming)
  • 14.00 – 16.00: $\,$ Group exercises in the study areas (b. 302, bottom floor)
  • 16.00 – 17.00: $\,$ Weekly test

Group Exercises

  1. Determination of Indefinite Integral. By Hand
  2. Divergence and Curl. By Hand
  3. Explosion, rotation og implosion fields
  4. Tangential Line Integrals of Gradient Vector Fields
  5. Circulations in the Plane
  6. Study of Divergence (Advanced)

Weekly Test

For all Weekly Tests, the following applies:

  • The test is an on-location test, meaning it can only be accessed in the study area.
  • No electronic aids are allowed (except for your own notes on e.g. a tablet).
  • The test can be accessed in the the Möbius quiz system via a link on DTU Learn in the module for 01006 (in the top menu click “Möbius”).
  • Your solutions to the test questions must be typed into Möbius without in-between calculations or steps. The result is automatically evaluated by Möbius.
  • To ensure a smooth experience use the Firefox or Chrome browser, and disable any add-blocker.
  • Use a DTU network.
  • You may discuss the test questions with fellow students in your study group, but you have your own version of the test with scrambled numbers that you yourself must solve and enter into Möbius.
  • During the final hour on Fridays you have one attempt. Passing this attempt will grant you 1 bonus point. From Friday at 18:00 until Wednesday at 18:00 the test is reopened for repeated attempts. Passing during this phase will grant you ½ bonus point.