**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