Calculus: Analytic Geometry and Calculus, with Vectors

10 Polar, cylindrical, and spherical coordinates 10.1 Geometry of coordinate systems We begin with a glimpse of a (or the) major ...

10.1 Geometry of coordinate systems 527 not enough todetermine the position ofa point P. It turns out to be convenient to determ ...

528 Polar, cylindrical, and spherical coordinates except in special situations where there is an explicit agreement to the contr ...

10.1 Geometry of coordinate systems 529 ±2, ,the numbers p, 4, + 2na constitutea set of polar coordinates of P. We could (and so ...

530 Polar, cylindrical, and spherical coordinates two of the formulas can be replaced by the single formula 4) = tan-' ylx. Thes ...

10.1 Geometry of coordinate systems 531 0 to a as 20 increases from 2Tr to 57r/2 and hence as ¢ increases from 7r to Sir/4. This ...

532 Figure 10.171 7/4, a2 cos 20 increases from 0 to a2 and then decreases to 0. This infor- mation enables us to sketch the two ...

10.1 Geometry of coordinate systems 533 2 With the aid of Figure 10.12, show that the formulas giving the cylindrical coordinate ...

534 Polar, cylindrical, and spherical coordinates 9 Sketch polar graphs of the equations (a)p=3+2cos¢ (b)p=3+4cos4> (c) p = a ...

10.1 Geometry of coordinate systems 535 and the formula (7.389) for curvature in rectangular coordinates, show that if p and ¢ a ...

536 Polar, cylindrical, and spherical coordinates This gives (2) and squaring gives (3) (2a - x)y = x x(2a - x), The graph of th ...

10.1 Geometry of coordinate systems 537 is the polar equation of the conchoid The graph consisting of the two solid branches nea ...

538 Polar, cylindrical, and spherical coordinates so 0 = 20 and the given angle AOP is 3q5. Thus the line OP1 trisects the given ...

10.2 Polar curves, tangents, and lengths 539 p > 0, so there can be no possible objection to use of Figure 10.21. intrinsic e ...

540 Polar, cylindrical, and spherical coordinates at a point P on it and, in particular, to have information about the angle ¢ ( ...

10.2 Polar curves, tangents, and lengths 541 The vector ul(t) is the unit vector in the direction of the projection of the vecto ...

542 Polar, cylindrical, and spherical coordinates For the case in which z(t) is identically zero and p(t) > 0, this gives the ...

10.2 Polar curves, tangents, and lengths 543 enable us to convert (7.26) into (10.281). For the special case in which 0(t) = t a ...

544 Polar, cylindrical, and spherical coordinates Problems 10.29 1 Obtain the standard polar equation of the conic K and use it ...

10.2 Polar curves, tangents, and lengths 545 are orthonormal vectors because they are unit vectors and their scalar (or dot) pro ...