LINE INTEGRALS #line #integrals #plane #curve #closed #orientation
Note :
* Curve In R2
is a function from [α
, β ] to R2.
* The
set {(x ,y)/x=f(t),y(t),t ε[α,β]} or the set { (x ,y)/x=t, y=f(t), tε[α,β]} is also called a curve in R2.
2. Curve
in R3 :
If ⲫ
, ψ , η are real valued functions of
t; then the set {(x , y, z) /x= ⲫ
(t), y=ψ(t),z= η(t), tε[α,β]} is called a curve in R3.
Note :
A curve in R3 is also a
function from [α , β ] to R3.
For
example the set {(x , y, z )/ x2+y2+z2=1} is a curve in R3.
Curve
in Rn
If ⲫ1,
ⲫ2,…, ⲫn are real valued functions of t, then the set
{( x1,x2,…,xn)/x1=
ⲫ1(t),x2= ⲫ2(t),…xn= ⲫn(t)/ tε[α,β]} is called a curve in Rn.
Note : Curve in Rn is a function from [α , β ]
to Rn.
Closed curve :
The set {(x ,y
)/ x= ⲫ (t), y=ψ(t),tε[α,β]} is said to be closed if ⲫ(α)= ⲫ(β) and ψ(α)= ψ(β).
Oriented curve :
If t
increases from α to β , the curve formed by
joining the corresponding points (ⲫ(t), ψ(t))
, in that order is said to be oriented from α
to β.
OR
An orientation of a
curve is choice of one of the two possible directions for travelling on the curve. In coordinate system, X-axis is oriented
towards the right and the Y-axis is in
upward orientation.
Line Integrals :
The following image tells about the derivation of a line integral.
Problems on Line Integrals :
1. Evaluate ∫ dx/(x+y), Where C is the curve x=at2,y=2at, 0≤t≤2.
2. 2. Show that ∫ xy dx =4/5,
where C is the arc of the parabola y2=x from (1,-1) to (1,1).
7. 7. Show that ∫x2 dx+xy dy=0, where C is the arc of the circle x2+y2=1 from (1,0) to (0,1).
10. Evaluate ∫(3xydx-y2dy), where C is the curve In the xy-plane y=2x2 from (0,0) to (1,2).
Y= a sin3t,
from (a,0) to (0,a).
13. Evaluate ∫(x2+y2)dx-2xydy, where C is the
rectangle x=0,x=a,y=0,y=b.
14. Evaluate ∫ (2a-y)dx-(a-y)dy, where
C is given by x=a(t-sint), y= a (1-cost).
# line #integrals #plane #curve #closed #orientation
#problems #parabola #circle #triangle
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