Cauchy-Riemann Equations Given the functions u(x, y) and v(x, y), verify that the Cauchy-Riemann equations
can be written in polar coordinate form as
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Calculus
- Mass-Spring System The mass in a mass-spring system see figure is pulled downward and then released, causing the system to oscillate according to x(t)=a1sint+a2cost where x is the displacement at time t,a1 and a2 are arbitrary constant, and is a fixed constant. Show that the set of all functions x(t) is a vector space.arrow_forwardHeat flux The heat flow vector field for conducting objects is F = -k∇T, where T(x, y, z) is the temperature in the object and k > 0 is a constant that depends on the material. Compute the outward flux of F across the following surfaces S for the given temperature distributions. Assume k = 1. T(x, y, z) = -ln (x2 + y2 + z2); S is the sphere x2 + y2 + z2 = a2.arrow_forwardCirculation and flux Find the circulation and the outward flux of the following vector fields for the curve r(t) = ⟨2 cos t, 2 sin t⟩ , for 0 ≤ t ≤ 2π. F = r/ | r | 2, where r = ⟨x, y⟩arrow_forward
- Torsion of a helix Compute the torsion of the helixr(t) = ⟨a cos t, a sin t, bt⟩, for t ≥ 0, a > 0, and b > 0.arrow_forwardHarmonic functions A scalar-valued function φ is harmonicon a region D if ∇2φ = ∇ ⋅ ∇φ = 0 at all points of D. Show that the potential function φ(x, y, z) = | r | -p is harmonicprovided p = 0 or p = 1, where r = ⟨x, y, z⟩ . To what vectorfields do these potentials correspond?arrow_forwardHeat flux The heat flow vector field for conducting objects is F = -k∇T, where T(x, y, z) is the temperature in the object and k > 0 is a constant that depends on the material. Compute the outward flux of F across the following surfaces S for the given temperature distributions. Assume k = 1. T(x, y, z) = 100e-x2 - y2 - z2; S is the sphere x2 + y2 + z2 = a2.arrow_forward
- Circulation and flux Consider the following vector field.a. Compute the circulation on the boundary of the region R (withcounterclockwise orientation).b. Compute the outward flux across the boundary of R. F = ⟨ -sin y, x cos y⟩ , where R is the square{(x, y): 0 ≤ x ≤ π/2, 0 ≤ y ≤ π/2}arrow_forwardCirculation and flux Find the circulation and the outward flux of the following vector fields for the curve r(t) = ⟨2 cos t, 2 sin t⟩ , for 0 ≤ t ≤ 2π. F = ⟨y - x, y⟩arrow_forwardLaplace equations Show that if w = ƒ(u, y) satisfies the Laplace equation ƒuu + ƒyy = 0 and if u = (x2 - y2) /2 and y = xy, then w satisfies the Laplace equation wxx + wyy = 0.arrow_forward
- Cylindrical coordinates Evaluate the Jacobian for the transformation from cylindrical coordinates (r, θ, Z) to rectangular coordinates (x, y, z): x = r cos θ, y = r sin θ, z = Z. Show that J(r, θ, Z) = r.arrow_forwardA. State the Fundamental Theorem of Calculus for Line Integrals. B. Let f(x, y, z) = x^2 + 2y^2 + 3z^2 and F = grad f. Find the line integral of F along the line C with parametric equations x = 1 + t, y = 1 + 2t, z = 1 + 3t, 0 ≤ t ≤ 1. You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.arrow_forwardCirculation in a plane A circle C in the plane x + y + z = 8 has a radius of 4 and center (2, 3, 3). Evaluate ∮C F ⋅ dr for F = ⟨0, -z, 2y⟩, where C has a counterclockwise orientation when viewed from above. Does the circulation depend on the radius of the circle? Does it depend on the location of the center of the circle?arrow_forward
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