Vector Functions and Space Curves In general a function is a rule that assigns to each element in the domain an element in the range A vector valued function or vector function is simply a function whose domain is a set of real numbers and whose range is a set of vectors We are most interested in vector functions r whose values are three dimensional vectors This means that for every number t in the domain of r there is a unique vector in V3 denoted by r t 1 Vector Functions and Space Curves If f t g t and h t are the components of the vector r t then f g and h are real valued functions called the component functions of r and we can write r t f t g t h t f t i g t j h t k We use the letter t to denote the independent variable because it represents time in most applications of vector functions 2 Example 1 Domain of a vector function If r t t 3 ln 3 t then the component functions are f t t 3 g t ln 3 t h t By our usual convention the domain of r consists of all values of t for which the expression for r t is defined The expressions t 3 ln 3 t and 3 t 0 and t 0 are all defined when Therefore the domain of r is the interval 0 3 3 Vector Functions and Space Curves The limit of a vector function r is defined by taking the limits of its component functions as follows Limits of vector functions obey the same rules as limits of real valued functions 4 Vector Functions and Space Curves A vector function r is continuous at a if In view of Definition 1 we see that r is continuous at a if and only if its component functions f g and h are continuous at a There is a close connection between continuous vector functions and space curves 5 Vector Functions and Space Curves Suppose that f g and h are continuous real valued functions on an interval I Then the set C of all points x y z in space where x f t y g t z h t and t varies throughout the interval I is called a space curve The equations in are called parametric equations of C and t is called a parameter We can think of C as being traced out by a moving particle whose position at time t is f t g t h t 6 Vector Functions and Space Curves If we now consider the vector function r t f t g t h t then r t is the position vector of the point P f t g t h t on C Thus any continuous vector function r defines a space curve C that is traced out by the tip of the moving vector r t as shown in Figure 1 C is traced out by the tip of a moving position vector r t Figure 1 7 Example 4 Sketching a helix Sketch the curve whose vector equation is r t cos t i sin t j t k Solution The parametric equations for this curve are x cos t y sin t z t Since x2 y2 cos2t sin2t 1 the curve must lie on the circular cylinder x2 y2 1 The point x y z lies directly above the point x y 0 which moves counterclockwise around the circle x2 y2 1 in the xy plane 8 Example 4 Solution cont d The projection of the curve onto the xy plane has vector equation r t cos t sin t 0 Since z t the curve spirals upward around the cylinder as t increases The curve shown in Figure 2 is called a helix Figure 2 9 Derivatives 10 Derivatives The derivative r of a vector function r is defined in much the same way as for real valued functions if this limit exists The geometric significance of this definition is shown in Figure 1 b The tangent vector r t a The secant vector Figure 1 11 Derivatives If the points P and Q have position vectors r t and r t h then represents the vector r t h r t which can therefore be regarded as a secant vector If h 0 the scalar multiple 1 h r t h r t has the same direction as r t h r t As h 0 it appears that this vector approaches a vector that lies on the tangent line For this reason the vector r t is called the tangent vector to the curve defined by r at the point P provided that r t exists and r t 0 12 Derivatives The tangent line to C at P is defined to be the line through P parallel to the tangent vector r t We will also have occasion to consider the unit tangent vector which is 13 Example 1 a Find the derivative of r t 1 t3 i te t j sin 2t k b Find the unit tangent vector at the point where t 0 Solution a According to Theorem 2 we differentiate each component of r r t 3t2i 1 t e t j 2 cos 2t k 14 Example 1 Solution cont d b Since r 0 i and r 0 j 2k the unit tangent vector at the point 1 0 0 is 15 Derivatives Just as for real valued functions the second derivative of a vector function r is the derivative of r that is r r For instance the second derivative of the function r t 2 cos t sin t t is r t 2 cos t sin t 0 16 Differentiation Rules 17 Differentiation Rules The next theorem shows that the differentiation formulas for real valued functions have their counterparts for vector valued functions 18 Example 4 Show that if r t c a constant then r t is orthogonal to r t for all t Solution Since r t r t r t 2 c2 and c2 is a constant Formula 4 of Theorem 3 gives 0 r t r t r t r t r t r t 2r t r t Thus r t r t 0 which says that r t is orthogonal to r t 19 Example 4 Solution cont d Geometrically this result says that if a curve lies on a sphere with center the origin then the tangent vector r t is always perpendicular to the position vector r t 20 Integrals 21 Integrals The definite integral of a continuous vector function r t can be defined in much the same way as for real valued functions except that the integral is a vector But then we can express the integral of r in terms of the integrals of its component functions f g and h as follows 22 Integrals and so This means that we can evaluate an integral of a vector function by integrating each component function 23 …
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