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sigh1342
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Homework Statement
Find $$\int_{C} z^3 ds $$ where C is the part of the curve $$ x^2+y^2+z^2=1,x+y=1$$ where$$ z ≥ 0 $$ then I let $$ x=t , y=1-t , z= \sqrt{2t-2t^2}$$ . Is it correct? Or there are some better idea?
Yes, that is correct. And, of course, [itex]ds= \sqrt{dx^2+ dy^2+ dz^2}= \sqrt{1+ 1+ (2- 2t)^2/(2t- 2t^2)} dt[/itex]. I thought about using the standard parameterization of the sphere and then adding the condition that x+y= 1, in order to avoid the square root, but that does not appear to give a simpler integral.sigh1342 said:Homework Statement
Find $$\int_{C} z^3 ds $$ where C is the part of the curve $$ x^2+y^2+z^2=1,x+y=1$$ where$$ z ≥ 0 $$ then I let $$ x=t , y=1-t , z= \sqrt{2t-2t^2}$$ . Is it correct? Or there are some better idea?
Homework Equations
The Attempt at a Solution
HallsofIvy said:Yes, that is correct. And, of course, [itex]ds= \sqrt{dx^2+ dy^2+ dz^2}= \sqrt{1+ 1+ (2- 2t)^2/(2t- 2t^2)} dt[/itex]. I thought about using the standard parameterization of the sphere and then adding the condition that x+y= 1, in order to avoid the square root, but that does not appear to give a simpler integral.
A path integral, also known as a line integral, is a mathematical concept used in physics and engineering to calculate the total value of a scalar or vector field along a given path or curve.
Parametric curves allow for a more flexible and efficient way of representing paths in path integrals. They allow for a single variable to represent both the position and direction of the path, making calculations easier and more accurate.
To calculate a path integral using a parametric curve, the curve is first broken down into small segments. The value of the scalar or vector field at each segment is then multiplied by the length of the segment and added together to find the total value along the curve.
Path integrals using parametric curves are used in a variety of fields, such as fluid dynamics, electromagnetism, and quantum mechanics. They are also used in engineering and computer graphics to model and analyze the behavior of complex systems.
While parametric curves offer many advantages in path integrals, they are not always the most accurate representation of a path. In some cases, a piecewise function or other methods may be more appropriate for calculating a path integral. Additionally, the complexity of the curve may make it difficult to find an appropriate parametric equation.