Chapter 17

August 18 2017

Chapter 17 Parametric Equations. Vectors in the Plane 17.1 Parametric Equations of Curves - when we think of a curve as the path of a moving point - it is often more convenient to study the curve by using two simple equations for x and y inter terms of a third independent variable t - x = f(t) and y = f(t) - than by using a single more complicated equation of the form - F(x, y) = 0 - this provides not only a description of the path on which the point moves - but also information about the direction of its motion and its location on the path for various values of t - the third variable in terms of which x and y are expressed is called a parameter - these equations are called parametric equations of the curve - if we want the rectangular equation of the curve again - we must eliminate the parameter from the equation - the use of parametric equations is very natural if we think of a curve as a path of a moving point whose position depends on time t - in motion problems - it is natural to use the time t as the parameter - in problems that are more concerned with geometry than physics - the most convenient parameter is likely to have some geometric significance - for instance - t as the slope, or angle - static equations can be parameterized - for example - parameterizing around the angle, slope, or tangent - the curve is traced out as the parameter increases from - inf to inf - our previous ways of representing curves - by rectangular coordinates and polar coordinates - are easy to fit into our present system of parametric representation - by using x or angle as the parameter 17.3 Vector Algebra. The Unit Vectors i and j - a physical quantity such as mass, temperature, or kinetic energy, is completely determined by a single real number that specifies its magnitude - these are called scalar quantities - in contrast - other entities called vector quantities posses both magnitude and direction - such as velocities, forces, and displacements - from the mathematical point of view - we don't merely represent a vector by a directed line segment - we say the vector is a directed line segment - this frees us to develop the algebra of vectors independently of any particular physical interpretation - two vectors are said to be equal - if they have the same length and direction - this enables us to move a vector from one position to another without changing it - we will discuss two algebraic operations on vectors - adding vectors - scalar multiplication - PR = PQ + QR - addition is commutative and associative - also can be found as the opposite vertex through the parallelogram rule - vector addition is well suited for working with forces in Physics - scalar multiplication - adjusts the size and direction of the vector - since the laws governing addition and scalar multiplication of vectors are identical with elementary algrebra - we are justified to using the familiar rules of algebra to solve linear equations involving vectors - a vector of length 1 is called a unit vector - if we divide any nonzero vector A by its own length - we obtain a unit vector A / |A| in the same direction - this simple fact is surprisingly useful - when we are working with vectors in the coordinate plane - it is often convenient to use the standard unit vectors i and j - placing any vector at the origin gives - A = a1 i + a2 j - where a1 is the x-component or the i-component - these components are scalars - so every vector in the plane is a linear combination of i and j - the value of this formula is based on the fact that such linear combinations can be manipulated by the ordinary rules of algrebra 17.4 Derivatives of Vector Functions. Velocity and Acceleration - we became acquainted with the algebra of vectors - now we will work with the calculus of vectors on problems of motion - when vectors and calculus are allowed to interact with each other - the result is a mathematical discipline of great power and efficiency for studying multi-dimensional problems of geometry and physics - this is vector calculus or vector analysis - we begin by point out the connection between vectors and the parametric equations of curves - suppose point P(x, y) moves along a curve in the xy-plane - and we know the position at anytime t - x = x(t), y = y(t) - these are the parametric equations for the path in terms of the time parameter t - R = OP = x(t) i + y(t) j - R is continuous and differentiable if x(t) and y(t) are - dR/dt = dx/dt i + dy/dt j - it is tangent to the path at the head of R - dR/dt has as its direction and length the direction and speed of our moving point - v = dR/dt, speed = |v| - a = dv/dt = d2R/dt2 - these concepts are direct extensions of one-dimensional motion - F = ma - the vector form of Newton's law shows the force and acceleration have the same direction - is is obvious by now the time t is a parameter of fundamental importance for studying the motion of a point P along a curved path - another important parameter is the arc length s - T = dR/ds - is a vector of unit length which is tangent to the curve at P - this is the unit tangent vector - v = dR/ds ds/dt = T ds/dt - the direction of velocity is given by the unit tangent vector T and its magnitude is given by ds/dt - our main aim in the next two sections is to obtain a corresponding formula for acceleration 17.5 Curvature and the Unit Normal Vector - we expressed the velocity v of our moving point P in terms of the unit tangent vector T - where T was obtained as the derivative of the position vector R with respect to arc length s - T = dR/ds - as a direst step toward the general accelerating formula - we must now analyze the derivative of T with respect to s - this requires us to examine the purely geometric concept of the "curvature" of a curve - the curvature at a point ought to measure how rapidly the direction of a curve is changing with respect to the distance along the curve - k = d angle / ds - for a circle, k = 1/r - this can tell us precisely how much the curve is bending - we can now analyze the derivative of the unit tangent vector T with respect to s - N = dT / d angle - which is the unit normal vector that is the negative reciprocal of the slope of T - dT / ds = dT / d angle d angle / ds = Nk - since T has constant length - only its direction changes as s varies 17.6 Tangential and Normal Components of Acceleration - consider a moving particle whose position at time t is given by the parametric equations - x = x(t) and y = y(t) - v = T ds/dt - where T is the unit tangent vector - this expression has physical meaning regardless of the choice of coordinate system - because T gives the direction of the motion and ds/dt gives its magnitude, the speed - to obtain a similar revealing expression for acceleration - a = T d2s/dt2 + Nk (ds/dt)2 - this is an important equation in mechanics - the vectors T and N serve as reference unit vectors much like i and j - they enable us to resolve the acceleration into two "natural" components - in the direction of the motion and normal to this direction - in contrast to the arbitrary components of i and j - the tangential component, d2s/dt2 - is simply the derivative of the speed of the particle along its path - the normal component has magnitude kv2 = v2/r - of course - the great importance of acceleration likes in the fact that when a particle of mass is acted on by a force F - it moves in accordance with Newton's second law of motion F = ma - the vectors F and a then have the same direction - and this fits with our intuitive understanding that when a force changes the direction of a moving particle - it pulls the particle away from the direction of the tangent toward the concave side of the path