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WOLFRAM NOTEBOOK
Slanted Line Integral
Slanted Line Integral
Generally speaking, the line integral of a function over a given curve is defined by selecting points on the curve, evaluating the function's values at these points, and then taking the limit of the evaluated expression as the number of selected points approaches infinity in an appropriate way.
The definition of the slanted line integral is of a different nature. The integration is performed over a domain on the left-hand side of the curve, which is bounded by the given curve and two lines which are parallel to the axes. Let be an integrable function and let be its cumulative distribution function, . Let be a given continuous and tendable curve (i.e., its tendency is defined for all its interior points). Suppose that is uniformly tended (i.e., its tendency indicator vector is constant for all its interior points). Then the slanted line integral of over is defined as follows:
f:
2
F:
2
F(x,y)=f(u,v)dudv
y
∫
v=0
x
∫
u=0
LN
LN
F
LN
(S)
∫
LN
∫∫
D
1
2
where is the positive domain of the curve (i.e., the domain bounded by and two lines that are parallel to the axes, such that is on the left of ), is the tendency of the uniformly tended curve , and and are the tendencies of the curve at the points and respectively. The letter in the notation of the slanted line integral stands for "Slanted". The curve should be a subcurve of another curve , to assure that the tendencies at the points and are well defined. In case the curve is not uniformly tended (the tendency indicator vector at the subcurve is not equal to the tendency indicator vector at the subcurve ), then the slanted line integral of over is defined as the sum: F≡F+F, where each of the integrals on the right is calculated as a slanted line integral over a uniformly tended curve.
D
LN
D
LN
τ(LN)
LN
τ(L)
τ(N)
L
N
"S"
LN
KLNO
L
N
LN
LM
MN
F
LN
(S)
∫
LN
(S)
∫
LM
(S)
∫
MN
In this Demonstration you can drag the points , , , , , flip the orientation of the curve, or vary the curvature of the subcurves and to see how these changes affect each of the parameters in the definition of the slanted line integral.
K
L
M
N
O
LM
MN