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Integration with variables Notes_Part_11

# Integration with variables Notes_Part_11 - z ph z Figure...

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ph squaresmallsolid z squaresmallsolid z Figure 7.18. Complex Curve and Tangent. notation x ( t ) = ( x ( t ) ,y ( t ) ) to complex notation z ( t ) = x ( t )+ i y ( t ). All the usual vectorial curve terminology — closed, simple (non-self intersecting), piecewise smooth, etc. — is employed without modification. In particular, the tangent vector to the curve can be identified as the complex number squaresmallsolid z ( t ) = squaresmallsolid x ( t ) + i squaresmallsolid y ( t ), where we use dots to indicated derivatives with respect to the parameter t . Smoothness of the curve is guaranteed by the requirement that squaresmallsolid z ( t ) negationslash = 0. Example 7.28. ( a ) The curve z ( t ) = e i t = cos t + i sin t, for 0 t 2 π, parametrizes the unit circle | z | = 1 in the complex plane. Its complex tangent squaresmallsolid z ( t ) = i e i t = i z ( t ) is obtained by rotating z through 90 . ( b ) The complex curve z ( t ) = cosh t + i sinh t = 1 + i 2 e t + 1 i 2 e t , −∞ <t< , parametrizes the right hand branch of the hyperbola Re z 2 = x 2 y 2 = 1. The complex tangent vector is squaresmallsolid z ( t ) = sinh t + i cosh t = i z ( t ). When we interpret the curve as the motion of a particle in the complex plane, so that z ( t ) is the position of the particle at time t , the tangent squaresmallsolid z ( t ) represents its instantaneous velocity. The modulus of the tangent, | squaresmallsolid z | = radicalbig squaresmallsolid x 2 + squaresmallsolid y 2 , indicates the particle’s speed, while its phase ph squaresmallsolid z measures the direction of motion, as prescribed by the angle that the curve makes with the horizontal; see Figure 7.18.

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Integration with variables Notes_Part_11 - z ph z Figure...

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