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1. Find the points on the ellipse n(t) ( (acost,bsint) = t ...

1. Find the points on the ellipse n(t) ( (acost,bsint) = t € [0,2m at which the curvature is minimized and maximized (with a &gt; b &gt; 0). 2. Show that the evolute of the ellipse n(t) = (acost, bsint) is the "affine astroid" Y(t) = a2 a 62 COS 62 - b a&...

1. (5 points) Let M. be a surface in R³, and suppose th...

1. (5 points) Let M. be a surface in R³, and suppose that at a given point PE M the principal curvature directions are: U1 = (3333 V7) with principle curvature K1=-2. 02 = 32(-1,1,0) with principle curvature K2 = 3. Find the normal curvature of M at p in the direction of U = 501 +22/202. 2...

2. Consider the ellipsoid of revolution r(A,B) : = (a cos Xc...

2. Consider the ellipsoid of revolution r(A,B) : = (a cos Xcos B,bsinBB with a &gt; b. (a) Consider three lines of constant latitude defined by B = TT4, 0 = TT4 and 4 = Find the total length of each line (i.e. from X = TT to \ = IT). Which one is smallest and which one is largest? (b) Sh...

(3) Let O : (a,b) R³ be a unit speed curve such that K(...

(3) Let O : (a,b) R³ be a unit speed curve such that K(s) &gt; 0 for all S and T (s) = T is a nonzero constant. Define = - - / B(o)do, (0 E (a,b)). . Namely, a (so) has the longest distance from the origin among all points on the curve. Prove that the curvature of O at so is at leas...

2. If you used the rule (6x, 6y) to transform the original f...

2. If you used the rule (6x, 6y) to transform the original figure into a new figure, how would the angles of the new figure compare to the angles in the original figure? How would the side lengths compare? 3. If you used the rule (1/2x, 1/2y) to transform the original figure, how would the angles c...

roblem 1. The Frenet frame of a curve in R³. For a regu...

roblem 1. The Frenet frame of a curve in R³. For a regular plane curve (and more generally for a regular curve on a 2-dimensional surface - e.g. the 2- sphere above) we could construct a unique adapted frame F. This is not the case for curves in higher dimensional spaces. Besides the curve b...

1.3.3. Check that re = using Te (z) = eid 2, (z)=z. 1.3.4. ...

1.3.3. Check that re = using Te (z) = eid 2, (z)=z. 1.3.4. Show that the complex form of a product of translations, rotations, and reflections is a function f(z) = ete , z or F(2) = cteir, z, where c € C, 0 € R. Show, conversely, that any function of this form is realizable as a produ...

Suppose that V11 (x) transforms like a covariant vector unde...

Suppose that V11 (x) transforms like a covariant vector under a general curvilinear coordinate transformation bc = x1V (x"). Show that the partial derivatives V11,V III ave Oxt themselves do not transform like a tensor but the combination Fuv = V11,U - Vv, 11 does transform like an an...

Question 1. Using the transformation law patterns for upper ...

Question 1. Using the transformation law patterns for upper and lower indices, predict a transfor- mation law giving Laß in terms of Lij in the presence of a coordinate transformation f : U U, and then show that your guess is correct. (This shows that the Lij functions do indeed form tensor...

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