1. THEORY: Scalar field (Definition, examples ). The 1 knd line integral ( properties, calculation, applications...

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1. THEORY: Scalar field (Definition, examples ). The 1" knd line integral ( properties, calculation, applications ). y sin' 4x dl where L: y= sin 4x, Task: Evaluate the integral i V1+16 cos? 4x 6. 2. Evaluate the complex line integral J(iz-2 Im z) dz , if L is a straight connecting the point z, =i with the point z, = 2+5i. 3. Find the angle between the gradients of the scalar fields v = and z. u = yx² +x²y² - z°x at the point M(1, 2, 1). 1. Find a flux of the vector field a = xi +3 yj + zk through a part o S: x+2y+z - 4 = 0 created as a result of intersection of the plar coordinate planes. %3D Using the Laplace transform, solve the differential equation, a the correctness of the obtained solution. y"+16y = 3, y(0) = 2, y'(0) = 0. eck out if the function f(z)= z² +|z|+2i is analytic, ar rivative in the case of existence. 1. THEORY: Scalar field (Definition, examples ). The 1" knd line integral ( properties, calculation, applications ). y sin' 4x dl where L: y= sin 4x, Task: Evaluate the integral i V1+16 cos? 4x 6. 2. Evaluate the complex line integral J(iz-2 Im z) dz , if L is a straight connecting the point z, =i with the point z, = 2+5i. 3. Find the angle between the gradients of the scalar fields v = and z. u = yx² +x²y² - z°x at the point M(1, 2, 1). 1. Find a flux of the vector field a = xi +3 yj + zk through a part o S: x+2y+z - 4 = 0 created as a result of intersection of the plar coordinate planes. %3D Using the Laplace transform, solve the differential equation, a the correctness of the obtained solution. y"+16y = 3, y(0) = 2, y'(0) = 0. eck out if the function f(z)= z² +|z|+2i is analytic, ar rivative in the case of existence.