## Vector Calculus And Differential Equations

### FormulaSheet 微分方程代写

Most of the formulas and theorems provided are stated without the conditions under which they apply. The notation used is the same as that used in lectures.

### Curve

For a given curve C in R3 with parametrization α : [a, b] → R3, ### LineIntegrals 微分方程代写 φ is normal to the tangent plane of the level surface {φ(x, y, z) = k} for a constant k. If F = ∇φ for some smooth φ, then F is conservative, φ is a potential function of F, and F · dr is path-independent.

### Curl 微分方程代写 If the domain of F is simply connected (irrelevant for the requirement of MATH2021) and ∇ × F 0, then F is conservative.

### Divergence ### Green’sFormula(Theorem) 微分方程代写 Using ”curl = ∇×”, ### DivergenceTheoremin2D ### SurfaceIntegral 微分方程代写

S is the graph of z f (x, y) for (x, y) in R. Then

Flux across S = ### TripleIntegral

In cylindrical coordinates:

∫∫∫ f (x, y, zdxdydz = ∫∫∫ f (r cos θ, r sin θ, zr drdθdz.

In spherical coordinates:

∫∫∫ f (x, y, zdxdydz = ∫∫∫ f  r cos θ sin ϕ, r sin θ sin ϕ, r cos ϕ  r2 sin ϕdrdθdϕ

### Gauss’Theorem:DivergenceTheoremin3D 微分方程代写 ### FourierSeries

For a 2L-periodic function f : R → R, the Fourier series of f : 微分方程代写 where the Fourier coefficients b0, (bn)n1 and (an)n1 are given by

### Instructions

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• Show all necessarywork, which will be marked.
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• Writeby hand on paper, and scan for online submission.
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1. #### Thefollowing questions are considered in the xy-plane, R2.

(a) Find the parametrisation α(s) for the unit circle S1= (x, yx2 + y2 = 1 , where s is the arc length parameter and the direction is counter-clockwise. Make sure to specify the range of s. 微分方程代写

(b) There is a very thin metal ring in the shape of S2= (x, yx2 + y2 = 4 . Under the parametrisation β(t) = (2 sin t, 2 cos t) for t  (   π, π], the density of it is 2 + sin t. Calculate the total mass of the ring.

(c) Usingthe definition of curve/line integral, calculate where S2 = {(x, y) | x2+y2 = 4} with the parametrisation β(t) = (2 sin t, 2 cos t) for t ∈ (−π, π] and the vector field F (x, y) = (y, x).

(d) Applying Greens’ Formula/Theorem, calculate the curve/line integral in (c) by translating it first to a double integral.

(e) Justify that the vector field G(x, y) = (x, y) is conservative, and the vectorfield F (x, y) = (y, x), already in (c), is not conservative.

(f) Using polar coordinates, calculate the double integral 微分方程代写

where D = {(x, y) | x2 + y2 ≤ 1}.

1. #### Thefollowing questions are considered in the xyz-space, R3.

(a) Consider the planex + 2y + 3z = 10 as a graph over the xy-plane. By surface integral, calculate the area of the part of the plane over the triangle in the xy-plane, with vertices (0, 0), (1, 0) and (0, 1).

(b) For the vector field F (x, y, z) = (xy2, yz2, zx2) and the unit sphere S = calculate the flux of F through S in the

outward direction by Gauss’ Theorem.

(c) For the same vector field F (x, y, z)  =  (xy2, yz2, zx2) as in (b),  calculate ∇ × F .

(d) For the vector field F (x, y, z) = (xy2, yz2, zx2), calculate the flux ofF through the unit lower hemisphere H = (x, y, z)  x2 +y2 +z2 = 1, z  0  in the direction which goes into the unit ball B = (x, y, zx2 + y2 + z2 1 by Stokes’ Theorem.

1. #### Thisquestion is on solving  微分方程代写

(a) Find the particular solution for the unknown y(x): • Find the particular solution for the unknown y(x):
• Find the particular solution for the unknown y(x): • Find the general solution for the unknown y(x): by the method of Variation of Parameters.

• Find the general solution for the unknown y(x): for x ∈ [−π, π] using Fourier series.

• Findthe particular solution for the unknown y(x): using Laplace Transform.  QQ在线咨询

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