b) In order to do the previous double integral, we want to use the substitutionx= uv-1, y=uvThe Jacobian of this transformation isJ(u,v)=The integral you found in part a) becomesu2 v2+ v um ) dv duu1 v1whereu1=u2=v1=v2=n=m= Suppose R is the region on the xy plane in the first quadrant determined by the inequalities9< xys 1093<10RLet C denote the boundary of the region R, oriented counterclockwise. Consider the vector fieldto3' 3F=a) Use Green's Theorem to write a double integral which computes the circulation (work)of F along the curve C:S. F-dr= JJ,Ody dx

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Let C denote the boundary of the region R, oriented counterclockwise. Consider the vector field F= 3 a) Use Green's Theorem to write a double integral which computes the circulation (work) of F along the curve C: So F•dr = dx

Let C denote the boundary of the region R, oriented counterclockwise. Consider the vector field F= 3 a) Use Green's Theorem to write a double integral which computes the circulation (work) of F along the curve C: So F•dr = dx

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

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:

Question

b) In order to do the previous double integral, we want to use the substitution
x= uv-1, y=uv
The Jacobian of this transformation is
J(u,v)=
The integral you found in part a) becomes
u2 v2
+ v um ) dv du
u1 v1
where
u1=
u2=
v1=
v2=
n=
m=

Suppose R is the region on the xy plane in the first quadrant determined by the inequalities
9< xys 10
93
<10
R
Let C denote the boundary of the region R, oriented counterclockwise. Consider the vector field
to
3' 3
F=
a) Use Green's Theorem to write a double integral which computes the circulation (work)
of F along the curve C:
S. F-dr= JJ,Ody dx

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