Chapter 9 Vector Calculus

Outline 9.1 Vector Functions 9.2 Motion on a Curve 9.3 Curvature and Components of Acceleration 9.4 Partial Derivatives 9.5 Directional Derivative 9.6 Tangent Planes and Normal Lines 9.7 Curl and Divergence 9.8 Line Integrals 9.9 Independence of the Path

Outline (cont’d.) 9.10 Double Integrals 9.11 Double Integrals in Polar Coordinates 9.12 Green’s Theorem 9.13 Surface Integrals 9.14 Stokes’ Theorem 9.15 Triple Integrals 9.16 Divergence Theorem 9.17 Change of Variables in Multiple Integrals

Vector Functions A vector function r has components that are functions of a parameter t For a given value of t, r is the position vector of a point P on a curve C

Vector Functions (cont’d.) If limits of the component functions exist, For all t for which the limit exists, If the components are differentiable, If the components are integrable,

Vector Functions (cont’d.) From the chain rule, where is a differentiable scalar function,

Motion on a Curve For a body moving along a curve, For circular motion in a plane For projectiles, position velocity acceleration position centripetal acceleration

Curvature and Components of Acceleration Curvature of C at a point is where The radius of curvature  is 1/  at a point P on a curve C is the radius of a circle that best fits the curve The circle is the circle of curvature Its center is the center of curvature

Partial Derivatives If the partial derivative with respect to x is and with respect to y is

Directional Derivative Gradient of a function is Consider

Tangent Plane and Normal Lines The normal line to a surface at P

Curl and Divergence Vector functions of two and three variables are also called vector fields

Line Integrals

Line Integrals (cont’d.)

Independence of Path A piecewise smooth curve C between an initial point A and a terminal point B is a path Line integrals involving a certain kind of field F depend not on the path both only on the endpoints

Independence of Path (cont’d.)

Double Integrals

Double Integrals (cont’d.) Type I Region Type II Region

Double Integrals in Polar Coordinates

Green’s Theorem The double integral over the region R bounded by a piecewise smooth curve C gives the area of the bounded region

Surface Integrals Surfaces integrals may be used to evaluate mass of a surface, flux through a surface, charge on a surface, etc.

Stokes’ Theorem Stokes’ Theorem is the three-dimensional form of Green’s Theorem

Stokes’ Theorem (cont’d.)

Triple Integrals Triple integral applications Volume of solids Mass of solids First and second moments of solids Coordinates of center of mass Centroid of solids

Divergence Theorem The Divergence Theorem is useful in the derivation of some of the famous equations in electricity and magnetism and hydrodynamics

Change of Variables in Multiple Integrals At times, it is either a matter of convenience or necessity to make a change of variable in a definite integral in order to evaluate it (to simplify the integrand or region of integration) The transformation of triple integrals follows suit