Math21A − 10/04 − Sug Woo ShinCurves and surfaces in the 3-D space.dim locally looks like parametrized by # of eqns in x,y,zCurve 1 line one variable twoSurface 2 plane two variables oneExample 0.1. (Locally looks like...)For a surface, think of a big sphere such as the earth. You are standing on the bigsphere. In your eyes, it’s just a huge plane, not a sphere. Similarly, if you walk alongthe equator of the earth, which is a curve, at each moment you would think that youare on a very long line, not a circle.Example 0.2. (Number of equations in x, y, z)We know that a sphere with center (a, b, c) and radius r is given by one equation(x − a)2+ (y − b)2+ (z − c)2= r2We also learned that the symmetric equation of a line has the formx − x0a=y − y0b=z − z0cSo a line is given by two equations (although the equation looks like one body, observethat it breaks up into two equalities.) The basic principle is that the dimension of yourobject gets smaller as you give more equations to be satisfied.Remark 0.3. (Short answer to why parametrization?)We will learn many examples of curves and surfaces in this course. It is very im-portant to parametrize them. First of all, it’s a useful way to “write down” curves andsurfaces. Once you have parametrization of curves or surfaces, “calculus”(anythingrelated to integration and differentiation) on curves and surfaces is possible. For in-stance, nice formulas for arc length or surface area are available. We’ll learn the arclength formula in the next class.1Some in-class exercises for section 12.1-12.2Exercise 0.4. ~r(t) = het2, e−t, t + 1i. Find the symmetric equation of the tangent lineat (e, e, 0).Exercise 0.5. ~r(0) = h3, 1, 2i,~r0(0) = h0, 1, 1i,~r00(t) = hcos t, et, 1i. Find~r0(t), andthen ~r(t).Exercise 0.6. Find all intersection points of the curve t 7→ h1, t, t2i with the plane−5x + 2y + z + 2 =
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