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1. Let γ : 􏰁 → 􏰁3 be a parametric curve with components f, g,h so γ(t) = (f (t), g(t),h(t)). Assume its component functions are infinitely differentiable. Determine which of the following statements are true or false. If false, give a counterexample. If true, explain why.

(1a) For all t ∈ 􏰁, the unit tangent vector T(t) is defined.

(1b) For all t ∈ 􏰁, the unit normal vector N(t), if it exists, satisfies N(t) = γ′′(t) . ||γ′′ ( t )||

(1c) For all t ∈ 􏰁, the unit binormal vector B(t) is uniquely defined provided T(t) and N(t) are both defined.

2. Factory wind tunnels are used to analyze the aerodynamic properties of fast moving objects like bikes, cars, trucks, planes, or spaceships. Large wind turbines combined with pulses of smoke create flow lines (see 0:20 to 0:40 of this video) that can be studied to perfect a design.

Let F : 􏰁4 → 􏰁3 be the time-dependent velocity vector field of the air flow in a wind tunnel. At time t ∈ 􏰁, the vector F(x, t) is the velocity of a particle in the air at point x ∈ 􏰁3 .

  1. (2a)  A pathline is the trajectory that individual fluid particles follow. The velocity of the particle will be determined by the velocity vector field at each moment in time. Write a formal statement that describes whenthetraceofγ:􏰁→􏰁3 isapathlineofF.
  2. (2b)  A streamline is the trajectory that individual fluid particles would follow if the flow were stable from a fixed moment in time and onwards. The velocity of the particle would be determined by the velocity vector field at that fixed moment in time. Write a formal statement that describes when the trace of γ:􏰁→􏰁3 isastreamlineofF.
  3. (2c)  Streamlines and pathlines are sometimes the same and sometimes not.
    • Do you expect them to be nearly the same in factory wind tunnels?
    • Do you expect them to be nearly the same close to a moving tornado? In a single full sentence each, explain why or why not.

3. A grayscale picture can be thought of as a function f : A → [0, 1] where A ⊆ 􏰁2 is a rectangular region and [0, 1] is the intensity. Higher values are brighter.

(3a) In a single full sentence, describe what the 1-level set of f represents. Use plain language.

  1. (3b)  You want to invert the grayscale intensity of your picture f . For example, black switches to white and white switches to black. Define a function g : A → [0, 1] in terms of f that would be the resulting picture after this inversion. No justification is required.
  2. (3c)  A colour version of your picture f can be described using the RGB colour model (see Numeric represen- tations). How can you think of your picture as a function h using RGB? State the domain and codomain of h and briefly explain what a value of your function h represents.

6. Like limit points, interior points have an equivalent sequential definition.

Lemma. Let S ⊆ 􏰁n be a set. Let x ∈ S. The point x is an interior point of S if and only if for any sequence {xk}k in 􏰁n converging to x, there exists K ∈ 􏰀 such that {xk}∞k=K ⊆ S.

You will study the proof the this lemma.
(6a) Here is a flawed proof of the “only if” direction.

One line has a critical flaw. Identify this line and briefly explain the issue.

1. Let S ⊆ 􏰁n and assume x ∈ S is an interior point.
2. Let {xk}k in 􏰁n be a sequence converging to x.
3. Since x is an interior point of S, there exists ε > 0 such that Bε(x) ⊆ S. 4. Sincexk →x,thereexistsK∈􏰀suchthat∥xK −x∥<ε.
5. Thus, {xk}∞k=K ⊆ S as required.

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数学代写 | MAT237 Multivariable Calculus with Proofs

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