Notice the tick marks pointing toward lower elevation. The figure above illustrates a depression and its representation using contour lines. (b) Notice how a mountain saddle, a ridge, a stream, a steep area, and a flat area are shown with contour lines. The figure above illustrates various topographic features. Contour lines that are very close together indicate a steep slope. Widely separated contour lines indicate a gentle slope. The peak is normally considered to be located at half the interval distance. Note: The intervals are increasing, therefore, the contours indicate a hill. From the contour map, a profile can be drawn of the terrain.Įxample 2 - Draw a profile showing the elevations of the contours. The even spacing indicates the hill has a uniform slope. The contour lines in this figure are equally spaced. Pick two contour lines that are next to each other and find the difference in associated numbers. Sharp contour points indicate pointed ridges.Įxample 1 - In the graphic below, what is the vertical distance between the contour lines? Contour lines tend to enclose the smallest areas on ridge tops, which are often narrow or very limited in spatial extent. A rounded contour indicates a flatter or wider drainage or spur. They then cross the stream and turn back along the opposite bank of the stream forming a "v". As a contour approaches a stream, canyon, or drainage area, the contour lines turn upstream. If the numbers associated with the contour lines are decreasing, there is a decrease in elevation. If the numbers associated with specific contour lines are increasing, the elevation of the terrain is also increasing. Index contours are bold or thicker lines that appear at every fifth contour line. A contour interval is the vertical distance or difference in elevation between contour lines. In order to learn more, you'll want to get a text, or take a class, or google around for scattered notes and videos on complex analysis (it's certainly possible to learn for free online).A contour line is a line drawn on a topographic map to indicate ground elevation or depression. $$f'(x)=\lim_z$, one uses "residue calculus," which is a part of the branch of mathematics called complex analysis (some sources call it complex variables too). With real functions $\Bbb R\to\Bbb R$, having a derivative There is a very important and special difference between $\Bbb R$ and $\Bbb C$ that occurs very soon when learning complex analysis. If you want to understand contour integrals, knowing about complex numbers is a must, so make sure you are familiar with them. Specifically, in the Bernoulli Number definition, how could I evaluate it by plugging in a value of $n$? Could you provide an example for when $n=0$ (in which $B_0 = 1$, as Wolfram says)Ĭontour integrals are integrals of complex-valued functions over a contour's worth of complex numbers in the complex plane $\Bbb C$, whereas line integrals are integrals of either scalar functions or vector-valued functions over a curve in $n$-dimensional space $\Bbb R^n$. Are they the same thing?Īlso, how would I evaluate a contour integral? One of the videos that actually DOES mention that symbol, it mentions something known as the Residue Theorem which also confuses me. I don't know what a Line Integral has to do with a Contour Integral. Whenever I look up a tutorial video for Contour Integrals, it directs me to Line Integrals, and nowhere in these videos do I see the integral symbol with a circle in the center of it. The Wikipedia article on Contour Integrals confuses me due to its wording. If anyone could tell me how, that would be great.) Looking at Bernoulli Numbers on Wolfram ( ) it defines Bernoulli Numbers using Contour Integrals (which unfortunately I do not know how to write in here. Recently, Bernoulli Numbers have caught my eye, for I am studying infinite series' and it is a part of the tangent function expanded as a Taylor Series. A very general question, I apologize, but as you read this, hopefully you get what I'm asking.
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