Curl and Divergence

Vector Field Let M and N be functions of two variables x and y defined on a plane region R. The function F defined by is called a vector field over R.

Definition: If is a vector field function then Divergence Definition: If is a vector field function then

Problems Find the divergence of the vector field (47-50,page no.1002) 48. 50. Find the divergence of the vector field at the indicated point(51-54 page no.1002) 54.

Definition: If is a vector field function then Curl Definition: If is a vector field function then

Problems Find the curl of the vector field at the indicated point (29-32 page no.1002) 32. 30. (2,-1,3)

Line Integral In mathematics, a line integral (sometimes called a path integral, contour integral, or curve integral; not to be confused with calculating arc length using integration) is an integral where the function to be integrated is evaluated along a curve

Definition If is defined in a region containing a smooth curve of finite length ,then the line integral of along is length of the longest subarc

Evaluation of line integral as definite Integral Given of plane curve C, we have that If then

Problems Problems(7-10,page 1014): 9.Evaluate the line integral over the specified region 5.(1008)Find the mass of a spring in the shape of the circular helix. Where the density of the wire

Line integral of a vector field Let be a continuous vector field defined on a smooth curve C given by .The line integral of on C is

Problems 33()1015: Find the work done by the force field on an object moving along the specified path 6(1010): Find the work done by the force field on particle as it moves along the helix from the point (1,0,0) to

Surface Integral In mathematics, a surface integral is a definite integral taken over a surface

It is now time to think about integrating functions over some surface, S, in three-dimensional space.  Let’s start off with a sketch of the surface S since the notation can get a little confusing once we get into it.  Here is a sketch of some surface S.

Area of a Parametric Surface If and

Where

Green’s Theorem

Statement Let be a simple connected region with a piecewise smooth boundary oriented counterclockwise (that is is traversed once so that the region always lies to the left).If and have continuous partial derivatives in an open region containing ,then

Derivation The following is a proof of the theorem for the simplified area D, a type I region where C2 and C4 are vertical lines. where C1 and C3 are curves.

Derivation y ie o x

again

Comparing we get Similarly we can use and Adding the above two integrals

Problems Ex-1(1028):Use Green’s theorem to evaluate the line integral Where C is the path from (0,0)to (1,1) along the graph of and from (1,1) to (0,0) along the graph Solution: Let

We Know that the Green’s theorem is y (1,1) O(0,0) x

Prob.2(1029): While subject to the force field a particle travels once around the circle of radius 3.Use Green’s theorem to find the work done by F. Solution: We know that Work=

Ex-7(1034):Use Green’s theorem to evaluate the line integral boundary of the region lying between and Solution: Let

Ex-12(1034): Evaluate boundary of the region bounded by between the graphs of Solution:

Ex-11(1034): Evaluate boundary of the region bounded by between the graphs of Solution:

The Divergence Theorem Let Q be a solid region bounded by a closed surface S oriented by a unit normal vector directed outward from Q. If is a vector field whose component functions have partial derivatives in Q, then

Problem Ex.1(1060): Let Q be the sold region bounded by the coordinate plane and the plane and let find Where S is the surface of Q. Solution: To evaluate we need four surfaces but by the divergence theorem we need only one triple integral

5(1065):Use the Divergence theorem to evaluate S is the surface bounded by the graphs of the equations and Similar problem :6(1065)

Stokes’s Theorem Stokes’s Theorem gives the relationship between a surface integral over an oriented surface S and a line integral along a closed space curve C forming the boundary of S.

In this section we are going to take a look at a theorem that is a higher dimensional version of Green’s Theorem.  In Green’s Theorem we related a line integral to a double integral over some region.  In this section we are going to relate a line integral to a surface integral.  However, before we give the theorem we first need to define the curve that we’re going to use in the line integral.

Let’s start off with the following surface with the indicated orientation. Around the edge of this surface we have a curve C.  This curve is called the boundary curve.  The orientation of the surface S will induce the positive orientation of C.  To get the positive orientation of C think of yourself as walking along the curve.  While you are walking along the curve if your head is pointing in the same direction as the unit normal vectors while the surface is on the left then you are walking in the positive direction on C.

Stokes’ Theorem Let S be an oriented smooth surface that is bounded by a simple, closed, smooth boundary curve C with positive orientation.  Also let   be a vector field then,

Problems Ex-1(1067): Let C be the oriented triangle lying in the plane evaluate Where Solution: Stokes’s Theorem is We know that

10.(1071): Evaluate

Final Part Divergence and Curl Line integral Surface Integral Green’s Theorem Gauss Divergence Theorem Stokes’s theorem