Jacobian Matrix Question?

Is the polar coordinates transformation conformal? Is the angle between v and w a 45 degree angle?

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  • Anonymous
    3 months ago
    Favourite answer

    EDIT. In reply to Winod's comment.(r, θ) is an ordered pair and represents position on a graph with r (1st coordinate) as the horizontal axis and θ (2nd coordinate) as the ‘vertical’ axis.The vector (1, 0) has a positive r component (1) and a zero θ component.  So on the graph it must point in the direction of r, i.e. to right.  But it is drawn pointing upwards.

    Also if you plot the xy graph (e.g. use an online graph-plotter) you will see clearly the angle is not 45º.

    END EDIT

    First note there is a mistake in the question.  It says:

    “(1, 0) is parallel to the line r = π”

    It should be (0,1).  The vector (green arrow, left diagram) points upwards, so it is (0,1), not (1, 0).

    I’ll use (1, 0) to be consistent with the text of the question, but note this makes the diagrams wrong:

    Green arrow on left diagram should point right.

    Green arrow on right diagram should point left.

    If you prefer, you could do the calculation using (0,1); this then matches the diagrams.

    _______________________

    The standard transformation matrix for polar coordinates is J(r,θ) =

    [cosθ -rsinθ]

    [sinθ  rcosθ]

    θ = π, r = π, cosθ = cosπ = -1, sinθ = sinπ = 0 gives:

    J(π,π) =

    [-1 0]

    [0 -π]

    ______________

    v =

    [-1 0][1]

    [0 -π][0]

    =

    [-1]

    [0]

    ________________

    w =

    [-1 0][1]

    [0 -π][1]

    =

    [-1]

    [-π]

    ________________

    v•w = (-1)(-1) + 0(-π) = 1 (dot product)

    We want the angle (A) between v and w.

    v is obviously has unit length, ||v|| = 1.

    ||w|| = √((-1)² + (-π)²) = √(1 + π²) =

    v•w = ||v||||w||cosA = 1√(1 + π²)cosA

    √(1 + π²)cosA = 1

    cos(A) = 1/√(1 + π²) = 0.303

    A = cos⁻¹(0.303) = 72º

    Note the angle shown on the diagram is 90º – 72º = 18º because of the mistake in the question.

    Anyway, this shows the transformation is not conformal.  But check my working.

  • Vaman
    Lv 7
    3 months ago

    Jacobian suggests coordinate transformation form one system to the other. It depends on scalar quantities. For example   dv=dxdydz can not be vector. Jacobian can not make it vector. Therefore. it is nothing to with the transformation that you are talking. Please verify. I am confused.

    The radial lines can not have vector sign, I do not nknow the meaning of pi here. If it is the angle, then wrt what?. dr and r dtheta can be perpendicular to one another. Therefore the area is r dr theta. Anticlock wise in the positive direction. May be dr (1,0) and rdtheta (0,1). They are perpendicular. If transfer them independently, they retain the same form. A is a vector, A' is a new vector, B is a vector.  B' is a new vector.If A dot B is 0 then A' dot B' also be 0. In that case, each is conformal. You need to talk to an expert. Individual components retain the conformal mapping.

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