代做MTH2004 Vector Calculus & Applications Term 2 2023-2024代做留学生SQL 程序

Vector Calculus & Applications

MTH2004

Term 2

2023-2024

Summative Coursework 1

Key:

• All questions are marked with a and are therefore home-work questions (summative assessments). Please complete these questions and hand them in by 12h00 Monday 26th February 2024 (via ELE).

• Answers should be handwritten and then scanned into PDF format (unless you have an ILP that specifies otherwise).

• The marks shown for each part of a question are only a guideline.

• Note that some marks will be awarded for mathematical com-munication (i.e. the correct use of clear, concise mathematical notation).

• Final totals will be expressed as percentages.

Exercise 1. Suppose that ϕ is a scalar field, and that u and v are vector fields. Prove the following identities using suffix notation:

(i) ∇ × (ϕv) = (∇ϕ) × v + ϕ(∇ × v)                                                    (3)

(ii) ∇ · (u × v) = v · (∇ × u) − u · (∇ × v)                                          (3)

(iii) ∇ × (u × v) = (∇ · v)u − (∇ · u)v + (v · ∇)u − (u · ∇)v                   (6)

(iv) (u · ∇)u = ∇ (2/1|u|2) − u × (∇ × u)                                            (8)

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Exercise 2. Calculation of integrals

(i) Let F be the vector field

and let P be the point (1, 1, 1). Consider two parameterised curves, C1 and C2, which join the origin to P:

Evaluate the two line integrals:

and show that they have the same numerical value. State why this is the case and verify the relevant condition for F.

Find a scalar potential f for which F = ∇f and show how the two integrals above could have been calculated by making use of f.    (10)

(ii) Let G be the vector field

Calculate the surface integral

where S is a surface consisting of the circular patch y 2 + z 2 = 2 in the plane x = 1. (You may assume that ˆn is the normal to this surface which points away from the origin.)   (8)

(iii) Calculate the volume integral

where r = |r| represents distance from the origin, n is a constant and V is the sphere of radius a centred on the origin.   (12)

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Figure 1: The vector field v, shown in a plane of constant z.

The applet at https://www.geogebra.org/m/QPE4PaDZ can be used to visualise 2D vector fields.

Exercise 3. The figure shows a visualisation of the vector field

Assume (as always) a right–handed coordinate system.

(i) Does the z-axis point out of the page or into the page. Explain your answer.      (1)

(ii) Find the divergence and curl of v.           (3)

(iii) In which regions of the domain shown is ∇ · v positive, negative and zero? Explain how these regions correspond to the pattern of vectors shown in the figure.             (3)

(iv) In which regions of the domain shown is (∇ × v)3 (i.e. the z–component of ∇ × v) positive, negative and zero? Explain how these regions correspond to the pattern of vectors shown in the figure.        (3)

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Exercise 4. Consider the relationship between Cartesian and spher-ical polar coordinates.

(i) With the aid of a sketch and simple geometry, show how each of the Cartesian coordinates (x, y, z) can be written as a function of the spherical polar coordinates (r, θ, ϕ). (As usual, r represents distance from the origin, ϕ represents ‘longitude’ and θ represents ‘co–latitude’.)       (5)

(ii) Hence, calculate all components of the Jacobian

(expressed as functions of r, θ and ϕ).            (5)

(iii) Hence, calculate all components of the Jacobian

(expressed as functions of r, θ and ϕ).             (5)

(iv) Hence, derive an expression for the volume element

dV = dx dy dz

(expressed as a function of r, θ, ϕ, dr, dθ and dϕ).             (5)

(v) With the aid of a sketch and simple geometry, explain how an alternative (geometric) argument can be used to derive the same expression for dV in spherical polar coordinates.          (5)

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Figure 2: Illustration of the right–hand rule. If the curled fingers of the right hand sit in the plane defined by u and v, pointing from u towards v (through the angle θ), then ˆt points in the direction of the thumb of the right hand.

Exercise 5. Given vectors u and v separated by an angle θ (mea-sured from u towards v), geometrical definitions of their dot– and vector–products are:

u · v = |u||v| cos θ                      (3)

u × v = |u||v|sin θ tˆ                   (4)

Here, ˆt is a unit vector, orthogonal to both u and v, which is obtained from the right–hand rule (see Figure).

(i) Show, by means of a sketch and simple geometry, that the compo-nent of v in the direction of u (in other words, the projection of v onto the line defined by u) is

(v · ˆu) ˆu                 (4)

(ii) Let v be a non–zero vector defined by the vector triple product

v = a × (b × c)                        (11)

Show, by means of a sketch and simple geometry, that v can be ob-tained by projecting a into the plane defined by b and c, and then rotating this projection within the plane by 90◦ (in the direction from c towards b).

[Hint: Express a as the sum of two vectors that lie, respectively, within and normal to the plane of interest.]

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