This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1912 Excerpt: ...of the directions AB and AC in the xy plane. We may call this ratio the scale of depiction of areas at the point A. dpdJ/ dtpd/ du dv dv du is called the functional determinant of the functions p(u, v) and $u, v). We have found the scale of depiction of lengths in the directions AB and AC. Let us now try to find it in ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1912 Excerpt: ...of the directions AB and AC in the xy plane. We may call this ratio the scale of depiction of areas at the point A. dpdJ/ dtpd/ du dv dv du is called the functional determinant of the functions p(u, v) and $u, v). We have found the scale of depiction of lengths in the directions AB and AC. Let us now try to find it in any direction whatever. From any point A in the uv plane, whose coordinates are u and v, we pass to a point D close by whose coordinates are u + Am, v + Av. In the xy plane we find the corresponding points A and D with coordinates (Fig. 51). Fiq. 51. The length of AD and the angle of its direction we denote by Ar and a in the uv plane and by As and X in the xy plane. The limit of the ratio As/Ar, to which it tends, when D approaches A without changing the direction AD is the scale of depiction at the point A in the direction AD. Writing Ax = (pu cos a + pv sin a)Ar + terms of higher order, Ay--cos a + pv sin a)Ar + terms of higher order. Dividing by Ar and letting Ar decrease indefinitely, we have in the limit dx-gr--pu cos a + pv sin a, These equations show the scale of depiction ds/dr corresponding to the different directions X in the x, y-plane and a in the u, v By introducing complex numbers we can show the connection still better. Let us denote z2 = Pv + iM, Multiplying the second of the two equations by i and adding both they may be written as one equation in the complex form: The modulus of z is the scale of depiction of the uv plane at the point A in the direction a. The angle of z gives the direction in the xy plane corresponding to the direction a. For a = 0 we have z = zi and for a = 90, z = Z2. Let us substitute plane. z = aeai + be. This suggests a simple geometrical construction of the complex numbers z for different values of a....
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