{"id":22678,"date":"2023-02-28T03:30:08","date_gmt":"2023-02-28T03:30:08","guid":{"rendered":"https:\/\/www.booksofall.com\/de\/?post_type=product&p=22678"},"modified":"2023-02-28T03:30:08","modified_gmt":"2023-02-28T03:30:08","slug":"clp-4-vector-calculus","status":"publish","type":"product","link":"https:\/\/www.booksofall.com\/de\/clp-4-vector-calculus\/","title":{"rendered":"CLP-4 Vector Calculus"},"content":{"rendered":"

Chapter 1 Curves\u00a0<\/b><\/h2>\n

We are now going to study vector-valued functions<\/a> of one real variable. That is, we are going to study functions that assign to each real number\u00a0t\u00a0<\/i>(typically in some interval) a vector1\u00a0r<\/b>(t<\/i>). For example<\/p>\n

r<\/b>(t<\/i>) = (\u00a0x<\/i>(t<\/i>), y<\/i>(t<\/i>), z<\/i>(t<\/i>))<\/p>\n

might be the position of a particle at time\u00a0t<\/i>. As\u00a0t\u00a0<\/i>varies,\u00a0r<\/b>(t<\/i>) sweeps out a curve.<\/p>\n

\"\"<\/p>\n

While in some applications\u00a0t\u00a0<\/i>will indeed be \u201ctime\u201d, it does not have to be. It can be simply a parameter that is used to label the different points on the curve that\u00a0r<\/b>(t<\/i>) sweeps out. We then say that\u00a0r<\/b>(t<\/i>) provides a parameterization<\/a> of the curve.<\/p>\n

Example 1.0.1 Parametrization of\u00a0<\/b>x<\/i>2 +\u00a0y<\/i>2 =\u00a0a<\/i>2.\u00a0<\/b>While we will often use\u00a0t\u00a0<\/i>as the parameter in a parametrized curve\u00a0r<\/b>(t<\/i>), there is no need to call it\u00a0t<\/i>. Sometimes it is natural to use a different name for the parameter. For example, consider the circle\u00a0x<\/i>2 +\u00a0y<\/i>2 =\u00a0a<\/i>2. It is natural to use the angle\u00a0\u03b8\u00a0<\/i>in the sketch below to label the point<\/p>\n

\"\"<\/p>\n

That is,<\/p>\n

r<\/b>(\u03b8<\/i>) = (\u00a0a\u00a0<\/i>cos\u00a0\u03b8 , a\u00a0<\/i>sin\u00a0\u03b8 <\/i>)\u00a0 \u00a0 \u00a0 \u00a0 0 \u2264 \u03b8 <\u00a0<\/i>2\u03c0<\/i><\/p>\n

is a parametrization of the circle\u00a0x<\/i>2 +y<\/i>2 =\u00a0a<\/i>2. Just looking at the figure above, it is clear that, as\u00a0\u03b8\u00a0<\/i>runs from 0 to 2\u03c0<\/i>,\u00a0r<\/b>(\u03b8<\/i>) traces out the full circle.<\/p>\n

However beware that just knowing that\u00a0r<\/b>(t<\/i>) lies on a specified curve does not guarantee that, as\u00a0t\u00a0<\/i>varies,\u00a0r<\/b>(t<\/i>) covers the entire curve. For example, as\u00a0t\u00a0<\/i>runs over the whole real line, 2\/\u03c0\u00a0<\/i>arctan(t<\/i>) runs over the interval (\u22121,\u00a0<\/i>1). For all\u00a0t<\/i>,<\/p>\n

\"\"<\/p>\n

is well-defined and obeys x<\/i>(t<\/i>)2 +\u00a0y<\/i>(t<\/i>)2 =\u00a0a<\/i>2. But this\u00a0r<\/b>(t<\/i>) does not cover the entire circle because\u00a0y<\/i>(t<\/i>) is always positive. \u0003<\/p>\n

Example 1.0.2 Parametrization of\u00a0<\/b>(x<\/i>\u2212h<\/i>)2 +(y<\/i>\u2212k<\/i>)2 =\u00a0a<\/i>2.\u00a0<\/b>We can tweak the parametrization of Example 1.0.1 to get a parametrization of the circle of radius\u00a0<\/a>a\u00a0<\/i>that is centred on (h, k<\/i>). One way to do so is to redraw the sketch of Example 1.0.1 with the circle translated so that its centre is at (h, k<\/i>).<\/p>\n

\"\"<\/p>\n

We see from the sketch that<\/p>\n

r<\/b>(\u03b8<\/i>) = (\u00a0h<\/i>+\u00a0a\u00a0<\/i>cos\u00a0\u03b8 , k\u00a0<\/i>+\u00a0a\u00a0<\/i>sin\u00a0\u03b8\u00a0<\/i>)\u00a0 \u00a0 \u00a0 0 \u2264 \u03b8 <\u00a0<\/i>2\u03c0<\/i><\/p>\n

is a parametrization of the circle (x<\/i>\u2212\u00a0h<\/i>)2 + (y\u00a0<\/i>\u2212\u00a0k<\/i>)2 =\u00a0a<\/i>2. A second way to come up with this parametrization is to observe that we can turn the trig identity<\/a> cos2\u00a0t<\/i>+sin2\u00a0t\u00a0<\/i>= 1 into the equation (x<\/i>\u2212h<\/i>)2+(y<\/i>\u2212k<\/i>)2 =\u00a0a<\/i>2 of the circle by<\/p>\n