Let C be the positively oriented square with vertices 0, 1, 1 + i, i. Evaluate using the definition of a contour integral. sin z dz

Q: Convert the integral to polar coordinates.

A: Polar coordinates Option B

Q: Let z=x+ iy with x and y are real. What is the imaginary part of sin (x + iy)

A:

Q: Find the slope of the tangent in the positive x-direction to the surface z = 3x - 6xy at the point…

A: We have to find the partial derivative with respect to x

Q: Identify the surface for ρ = 4 cos φ

A: The given curve is: ρ=4cosφ Multiplying both the sides by ρ: ρ2=4ρcos φ Now, converting it into…

Q: Find the Jacobian for the change of variables defined by x = r cos θ and y = r sin θ.

A: To calculate the Jacobian for the change of variables, we need to calculate the following…

Q: 28. fc[(y – 2) dx + (z – x) dy + (x – y) dz]; C is the curve of intersection of x² + y? = 1, and a +…

A: The stokes theorem states that ∮CFdr=∬D∇×Fds. The given integral is: ∮Cy-zdx+z-xdy+x-ydz, where C is…

Q: (b) Evaluate the line integral dz along the simple closed contour C shown in the diagram. 1

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Q: Evaluate  ∫C (3y + exsinx) dx + (8x - ln(y3+2)) dy, where C is the circle x2 + y2 = 9  with positive…

A:

Q: Evaluate | exp(-2²)dx, using an iterated integral and polar coordinates.

A: Using Cartesian coordinates to polar coordinates.

Q: circle centered d complex varia tour integral egration is take

A:

Q: Evaluate [F.dr, where C is the circle x² + y² = 4 oriented counterclockwise and F = (y,-x). Set up…

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Q: Determine the y-coordinate of the centroid of the shaded area. x = ky? I---x

A: The given problem is to find the y coordinate of the centroid of the given shaded region. Shaded…

Q: Find the points of intersection between r =2 cos20 andr=- %3D 1- cose

A: To find the intersection point two curve we have to find r,θ by solving given curve.

Q: Identify the y coordinate of the centroid of the shaded area in terms of r₁ r2, and a. a x

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Q: 9. Use the Residues Theorem to calculate 1+ sin² ª -de. Sketch the contour you used.

A: We have to find integral using Residue Theorem.

Q: Differentiate, T. (cb-da-dbT btd with respect to T

A: