Lecture 13: Vector Calculus III 1 Key points Line integrals (curvilinear integrals) of scalar fields Line integrals (curvilinear integrals) of vector fields Surface integrals Maple int PathInt LineInt 2 Line integrals of scalar field The integral of a scalar field along a path is written as where ds is infinitesimal segment along the path There is another kind of the line integral of scalar fields where dr is the element of the line along the path In the Cartesian coordinates, it can be expressed as Note that the result is a vector This type of integrals is not common in physics We discuss only the first kind Example in physics The length of path: The time of travel: For simplicity, we study only two dimensional cases in this lecture Path given as y=y(x)
The line segment can be written as Then, we have the line integral Example Consider a projectile motion where a and b are x and y component of the initial velocity, respectively Find the length of the path = = Path given as a parametric function [x(s),y(s)] Using and, Then, we have the line integral Example 21 Consider a projectile motion Find the length of the path from t=0 to t=2b/g = = This answer is the same as the previous example as it should be Example 22
Evaluate the integral along the path specified by,, = 2 3 3 Line integrals for vector fields: Type 1 There are two types of line integrals or curvilinear integrals of a vector field along a path: and where d is the element of line, an infinitesimal vector tangent to the path In the Cartesian coordinates The result of the first integral is scalar while the second one is vector We discuss the first kind in this section Expressing the vector field also in the Cartesian coordinates, Then, the line integral is The first integral in the right hand side the value of y and z must be specified Similarly, in the second integral the value of x and y must be given Hence, this integral does not have a unique value unless the relation among x, y and z is given The relation does not have to be one-to-one relation For simplicity, we study only two dimensional cases in this lecture Examples in physics (1) Work (2) Scalar potential (3) Ampere theorem Path consisting of multiple line segments Consider a vector field We integrate it along a path starting at (0,0) and
ending at (1,1) We take a path consisting of two segments of straight lines: a horizontal segment from (0,0) to (1,0) and a vertical segment from (1,0) to (1,1) Than means when x is varied, y=0 and when y is varied, x=1 Then, we have = = 2 = Path given as a function y=y(x) Using we can eliminate y
Example Consider a vector field (0,0) to (1,1) That means along y=x again We integrate it along a straight line from = = 3 2
Example -3D = 1 2 Parametric path [x(t),y(t)] Using and,
Example 1 Consider a vector field arc of a circle centered at (1,0) The path is given by and once more This time, we integrate it along an = (3411) (3412)
Example 2 - Closed loop Integrate the vector field along a circle of radius 1 centered at (0,0), =
= Line integral of Gradient Consider a line integral of gradient of a scalar field along a path starting from to Example: Work and potential energy Work is defined as If the force is conservative, Hence, path,respectively, where and are starting and ending points of the
4 Line integrals for vector fields: Type 2 For the second kind of the line integral of a vector field is Since it is a cross product of two vectors the result is a vector defined by Examples in physics Biot-Savart law: The magnetic field due to a steady current along a line is given by where is the element of length along the line of the current and is the position vector measured from the element of the line to the observation point A loop of wire carrying a current is placed in a magnetic field The force exerted on the wire by the magnetic field is given by Example Consider a vector field = = Evaluate for the circular path on the xy plane with radius Using parametric representation of the path:, (because ) =0 (because ) =0 (because ) (because ) = 0
Hence the integral is zero vector This result is obvious because along the circle both and are on the xy plane and their cross product is zero = 0 5 Surface integrals The surface integral of scalar fields is written as, and the surface integral of vector fields as where rule atomic is the surface element whose direction is normal to the surface determined by right-hand In physics, the second form is by far the most common form of the surface integral Examples in physics Gauss's law: S) Example Consider a unit cube centered at the origin Its edges are parallel to the axes Find where S is all faces of the cube There are six faces Chose the outward direction is positive For the face at, Due to the cubic symmetry, all faces have the same integral value Hence, Homework: Due 10/17, 11am 131 Consider a projectile motion Find the time of travel from x=0 to x=r
132 1 2 Evaluate Consider a force field Calculate the work done by the force in moving from (1,1) to (3,3) along two different paths (You pick two paths) 3 Consider a force field Calculate the work done by the force along a complete 133 unit circle defined by Evaluate over the whole surface of the cylinder bounded by