At Q(1, 1), f>0, 8, > 0, and V· F>0. F = r?, y) Q(1, 1) V.F>0. for x>- V.F<0 for x<- 2 Figure 17.39
Arc Length
Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length.
Line Integral
A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.
Triple Integral
Examples:
Verify the claim made in the given problem below section d by showing that the net outward flux of F across C is positive.
(Hint: If you use Green’s Theorem to evaluate the integral ∫C ƒ dy - g dx,
convert to polar coordinates.)
Divergence from a graph To gain some intuition about the divergence,
consider the two-dimensional vector field F = ⟨ƒ, g⟩ = ⟨x2, y⟩ and a circle C of radius 2 centered at the origin (see figure).
a. Without computing it, determine whether the two-dimensional divergence is positive or negative at the point Q(1, 1). Why?
b. Confirm your conjecture in part (a) by computing the two-dimensional divergence at Q. c. Based on part (b), over what regions within the circle is the divergence positive and over what regions within the circle is the divergence negative?
d. By inspection of the figure, on what part of the circle is the flux across the boundary outward? Is the net flux out of the circle positive or negative?
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