This seems like a complex set of mathematical problems involving vector calculus and parametric equations. Let's break it down:

(i) We need to show that a · a' = b · b' = c · c' = 1.

Given that a' = b × c(a, b, c), b' = c × a(a, b, c), c' = a × b(a, b, c), and (a, b, c) = a · (b × c), we can substitute these into the equations.

For a · a', we get a · (b × c(a, b, c)) = a · (b × c) = 1, because the dot product of a vector with the cross product of two other vectors is equal to the volume of the parallelepiped formed by the three vectors, which is 1 in this case.

Similarly, for b · b' and c · c', we get b · (c × a(a, b, c)) = 1 and c · (a × b(a, b, c)) = 1 respectively.

(ii) We need to show that a' · b = a' · c = 0, b' · a = b' · c = 0 and c' · a = c' · b = 0.

This can be shown by the fact that the dot product of two perpendicular vectors is zero. Since a', b', and c' are defined as the cross products of the other two vectors, they are perpendicular to the vectors being crossed, hence their dot product is zero.

2. (a) We need to prove the identity ∇ × (φA) = (∇φ) × A + φ(∇ × A).

This is a standard result in vector calculus known as the product rule for the curl of a vector field. It can be proven by applying the definition of the curl and the product rule for derivatives.

(b) We need to find a parametric equation for the semicircle x² + y² = a², y 0, using as a parameter the slope t = dy/dx of the tangent to the curve at (x, y).

The equation of the semicircle can be parametrized as x = a cos θ, y = a sin θ, where θ is the angle made by the radius with the x-axis. The slope of the tangent to the curve at any point is given by dy/dx = -cot θ = -cos θ/sin θ = -x/y. We can use this to express θ in terms of t and substitute back into the parametric equations to get them in terms of t.

Question

This seems like a complex set of mathematical problems involving vector calculus and parametric equations. Let's break it down:

(i) We need to show that a · a' = b · b' = c · c' = 1.

Given that a' = b × c(a, b, c), b' = c × a(a, b, c), c' = a × b(a, b, c), and (a, b, c) = a · (b × c), we can substitute these into the equations.

For a · a', we get a · (b × c(a, b, c)) = a · (b × c) = 1, because the dot product of a vector with the cross product of two other vectors is equal to the volume of the parallelepiped formed by the three vectors, which is 1 in this case.

Similarly, for b · b' and c · c', we get b · (c × a(a, b, c)) = 1 and c · (a × b(a, b, c)) = 1 respectively.

(ii) We need to show that a' · b = a' · c = 0, b' · a = b' · c = 0 and c' · a = c' · b = 0.

This can be shown by the fact that the dot product of two perpendicular vectors is zero. Since a', b', and c' are defined as the cross products of the other two vectors, they are perpendicular to the vectors being crossed, hence their dot product is zero.

2. (a) We need to prove the identity ∇ × (φA) = (∇φ) × A + φ(∇ × A).

This is a standard result in vector calculus known as the product rule for the curl of a vector field. It can be proven by applying the definition of the curl and the product rule for derivatives.

(b) We need to find a parametric equation for the semicircle x² + y² = a², y > 0, using as a parameter the slope t = dy/dx of the tangent to the curve at (x, y).

The equation of the semicircle can be parametrized as x = a cos θ, y = a sin θ, where θ is the angle made by the radius with the x-axis. The slope of the tangent to the curve at any point is given by dy/dx = -cot θ = -cos θ/sin θ = -x/y. We can use this to express θ in terms of t and substitute back into the parametric equations to get them in terms of t.

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