Math 275 Notes (Ultman) Topic 5.9: Divergence and The Divergence Theorem Textbook ection: 16.9 From the Toolbox (what you need from previous classes): Computing partial derivatives. Computing the dot product. etting up and evaluate triple integrals. Knowing what a vector surface integral (aka flux integral) is. etting up and evaluating vector surface integrals. Related ideas: the gradient of a function f ; Green s Theorem. Learning Objectives (New kills) & Important Concepts Learning Objectives (New kills): Compute the divergence of a vector field. Apply the Divergence Theorem to evaluate surface integrals. Recognize when the Divergence Theorem can be used. Important Concepts: uppose is a closed surface, oriented outwards, that bounds a 3-dimensional region. The Divergence Theorem relates the flux (surface) integral of a vector field F out of, to a triple integral over : F d = divf dv. divf = F measures the local expansion/contraction of the vector field F.
The Big Picture If is a closed surface, oriented outward, and is the 3-dimensional region bounded by, the Divergence Theorem states that the flux (vector surface) integral of a vector field F out of is equal to the triple integral of the divergence of F over : F d = divf dv. The right-hand side of this equation is a triple integral, which we learned to evaluate earlier this semester. Divergence measures the local expansion/contraction of the vector field F (the net flow of F moving towards or away from a point). Divergence multiplied by the volume element measures the flux of F out of an infinitesimal box of volume dv : divf dv = F d. infinitesimal box o the Divergence Theorem states that adding up the flux of F out through infinitesimally small boxes through the region is the same as the total flux (vector surface integral) out through the boundary surface =. How to Read the Divergence Theorem Given a vector field F (x, y, z) = P (x, y, z) î + Q(x, y, z) ĵ + R(x, y, z) ˆk, the Divergence Theorem equates a vector surface integral and a triple integral: F d = divf dv Type of Integral: Vector urface Integral Triple Integral (2-d) (3-d) Domain of Integration: a closed surface in R 3 the 3-d region inside Integrand: F (x, y, z) divf = F = P x + Q y + R z 2
More Details The Nabla (or Del) Operator In Cartesian coordinates, the nabla operator is a vector whose components are partial derivatives: = y ĵ + z ˆk e have already seen the nabla operator applied to compute the gradient of a function: [ f = y ĵ + ] z ˆk f = x f î + y f ĵ + z f ˆk = f f y ĵ + f z ˆk Computing Divergence The divergence of a vector field in Cartesian coordinates can be computed by taking the dot product of the nabla operator and the field: F (x, y, z) = P (x, y, z) î + Q(x, y, z) ĵ + R(x, y, z) ˆk divf = F [ = y ĵ + ] z ˆk = P (x, y, z) + Q(x, y, z) + R(x, y, z) x y z = P x + Q y + R z [ ] P (x, y, z) î + Q(x, y, z) ĵ + R(x, y, z) ˆk Note: f turns a scalar-valued function f into a vector field, but divf turns a vector field into a scalar-valued function! hat Divergence Measures Divergence can be interpreted as the (local) expansion/contraction of a vector field. Imagine an infinitesimal box around each point in the domain of F : 3
If divf > 0 at a point, the net flow of F through the box is positive, and the field is expanding at that point. If divf < 0 at a point, the net flow of F through the box is negative, and the field is contracting at that point. A field F with divf = 0 at all points is sometimes called incompressible or divergence free. Geometric motivation for the Divergence Theorem divf dv measures the flux of the field F out through an infinitesimal box of volume dv : divf dv = F d. infinitesimal box The triple integral adds up these infinitesimal fluxes. Notation Boxes on the interior of the region share faces with their neighbors. The flux of the field F through these shared faces have the same absolute value, but opposite signs, so they cancel. Faces of boxes along the bounding surface = don t have a neighbor to cancel with. Adding all these up leads to the total flux out through the bounding surface =. If a surface is closed, the double integral symbol may include a circle:. It is also ok to use the generic notation. If is a 3-d region, its boundary may be denoted by. This boundary is a surface, so you may see either or used to denote the boundary surface of in the Divergence Theorem: F d = divf dv or F d = divf dv 4
Comparison of the Divergence Theorem, Green s Theorem, FTLI, FTC ˆ b FTC: f (b) f (a) = f (t) dt a ˆ FTLI: f (Q) f (P ) = f dr C Q Green s: F dr = C D x P y da Divergence: F d = divf dv Technical Conditions In order for Green s Theorem to hold, there are certain conditions that need to be met. In this class, we will only stress that the surface be closed with outward orientation; you may encounter the additional conditions in future classes. urface: = is closed and oriented outwards. It forms the boundary of a 3-dimensional region. Field: Both the surface and the interior region are contained in an open set (a bubble around the region and it s boundary surface = ) throughout which the field F is defined and divf exists and is and continuous. 5