Chapter 3 Vector Functions 3. Vector Functions and Space Curves efinition. A vector-valued function of one variable is a function whose domain is a set of real numbers and whose range is a set of vectors, usuall in R 3. If r :! R 3 is a vector-valued function then each component of r(t) is a function of t: r(t) =hf(t), g(t), h(t)i = f(t)i + g(t)j + h(t)k We call f(t), g(t) and h(t) thecomponent functions. ample. Let r(t) = t, 3sin(3t +3 /), p t 5. Find the domain of r(t), and find the range of each component function. Solution: As usual, the domain is implied to be all the real numbers that make the formulas defined. The onl component function that s not defined for some values of t is the square root. Thus, : t. We ll describe the range component b component: (, 3 apple apple 3, 5). efinition. If r(t) =hf(t), g(t), h(t)i then lim r(t) = lim f(t), lim g(t), lim h(t) t!a t!a t!a t!a ample. Let r(t) = Solution: t, 3sin(3t +3 /), p t 5 (if these limits eist).findlim t! r(t). lim r(t) = lim t, lim 3sin(3t +3 /), lim (p t 5) t! t! t! t! = h, 3, 5i 47
CHAPTR 3. VCTOR FUNCTIONS 48 efinition. If r(t) = hf(t), g(t), h(t)i and lim t!a r(t) =r(a), then r(t) iscontinuous at t = a efinition. The range of a vector-valued function r(t) is called a space curve. In other words, a space curve equals the set of all (,, ) such that = f(t) = g(t) = h(t) (3.) and as t varies. We call these equations the parametric equations of the curve, and call t the parameter. ample 3. Use a computer package to graph the space curve defined b r(t) = t, 3 cos(3t +3 /), p t 5. Solution: We show below the Matlab code, together with the resulting graph 3 4 4 5 5 t= linspace(,4*pi); = t; = 3*cos(3*t +3*pi/); = sqrt(t)-5; plot3(,,); label(''),label(''), label('') ample 4. Find the parametric equations of the following space curve, and use a computer package to graph the result Solution: The parametric equations are Notice that r(t) = cos(t)i + tj +sin(t)k = cos(t), = t, =sin(t), + = cos (t)+sin (t) = What this means is that all the (, )-values lie on the clinder + =. As the -value increases, the (, )-values wind around the clinder. We show below the Matlab code, together with the resulting graph
CHAPTR 3. VCTOR FUNCTIONS 49.5.5 5 5.5.5 t = linspace(,4*pi); = cos(*t); = t; = sin(*t); plot3(,,); label(''),label('' ),label('') ample 5. Find a vector function that represents the curve of intersection of the following shapes: + + =(sphere), = (plane). Solution: To intersect two surfaces we combine the two equations in the usual fashion, in this case substituting = into the other equation +( )+ = + = + = In the -plane this is an ellipse (although in the = plane the curve is a circle). The parametric equations of an ellipse start with sine and cosine, multipling one of these b a larger number to give the longer ais: = p cos(t), =sin(t), =sint, apple t apple. Shown below is the Matlab code and resulting graph.5.5.5.5 t = linspace(,*pi); = sqrt()*cos(t); = sin(t); = sin(t); plot3(,,); label(''),label('' ),label('')
CHAPTR 3. VCTOR FUNCTIONS 5 ample 6. [#36] Find a vector function that represents the curve of intersection of the following two shapes + =4, and = This is where we ended on Wednesda, Januar 3 Solution: The shape + =4iscircleinR, and so makes a vertical clinder in R 3. The parametric equations of a circle start with sine and cosine, and multipl these b the radius. So, in the -plane we get = cos t, =sint, t [, ] Now we combine the parametric equations with the =, and simplif the result with atrigidentit: = = 4 cos(t)sin(t) So now we can write, and in parametric equations: 9 = cos(t) =sin(t) >= =4sin(t) cos(t) >; r(t) = cos ti +sintj +sin(t)k apple t apple Here is the Matlab code and the resulting graph: t = linspace(,*pi); = *cos(t); = *sin(t); = 4*sin(t).*cos(t); plot3(,,); label(''),label('' ),label('') The above graphic shows the vector function that we found above, that is the intersection of two surfaces. But it doesn t show the surfaces themselves. The graphic below shows the original two surfaces themselves:
CHAPTR 3. VCTOR FUNCTIONS 5 t = linspace(,*pi); = *cos(t); = *sin(t); = 4*sin(t).*cos(t); plot3(,,, 'k', ' LineWidth', 3); hold on [c,c,c]=clinder(,5) surf(c,c,3*c) surf(c,c,-3*c) alpha(.3) [u,v]=meshgrid( linspace(-,)); =u.*v; mesh(u,v,) label(''),label(''),label('' ) hold off