1. THEORY: Definition of simply-connected and multiply-connected region. Cauchy's theorem for a simply-connected region, and its...

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1. THEORY: Definition of simply-connected and multiply-connected region. Cauchy's theorem for a simply-connected region, and its consequence theorem. Cauchy's integral formula (theorem, its consequence, application ). Task: Evaluate the contour integral zdz where (z +1)(z+2)(z+3)' C: z-1=2.5, using the Cauchy's integral formula. 2. Find a flux of the vector field ä = 2zi +(x- z)j +(2y+z)k through a part of the surface S: x-2y+ 2z -2=0 created as a result of intersection of the plane S with coordinate planes. 3. Represent the complex number z 2-2/3i graphically in the complex plane, find its principle value of argument and modulus. Calculate Vz, and represent its values graphically. 4. Find a directional derivative of the scalar field u(x, y, z) = at the point M(2, 2, 1) along the direction of the vector MA, if A(1, -1, 3). 5. Using the Laplace transform, solve the differential equation, and check the correctness of the obtained solution. y"+9y=e', y(0) = 0, y'(0) = 1. 6. Find a work of the vector field F = (2y-x)i-(y+x²)j on shifting a unit mass along a broken line ABC, if A(0, 2), B(2, 2) and C(3, 0). 1. THEORY: Definition of simply-connected and multiply-connected region. Cauchy's theorem for a simply-connected region, and its consequence theorem. Cauchy's integral formula (theorem, its consequence, application ). Task: Evaluate the contour integral zdz where (z +1)(z+2)(z+3)' C: z-1=2.5, using the Cauchy's integral formula. 2. Find a flux of the vector field ä = 2zi +(x- z)j +(2y+z)k through a part of the surface S: x-2y+ 2z -2=0 created as a result of intersection of the plane S with coordinate planes. 3. Represent the complex number z 2-2/3i graphically in the complex plane, find its principle value of argument and modulus. Calculate Vz, and represent its values graphically. 4. Find a directional derivative of the scalar field u(x, y, z) = at the point M(2, 2, 1) along the direction of the vector MA, if A(1, -1, 3). 5. Using the Laplace transform, solve the differential equation, and check the correctness of the obtained solution. y"+9y=e', y(0) = 0, y'(0) = 1. 6. Find a work of the vector field F = (2y-x)i-(y+x²)j on shifting a unit mass along a broken line ABC, if A(0, 2), B(2, 2) and C(3, 0).