14Chapter 6 Vector analysis Mathematical methods in the physical sciences 3rd edition Mary L. BoasChapter 6 Vector analysisLecture 19 Directional derivative; Gradient
155. Fields (장)Field: region + the value of physical quantity in the regionex) electric field, gravitational field, magnetic field
166. Directional derivative: gradient (방향 도함수 ; 기울기벡터) The change of temperature with distance depends on the direction. directional derivative1) definition of directional derivative(directional derivative for u: directional unit vector)
29Example 2. Find the value of path 1 (polar coordinate ) r=1 (constant)so, only d may be considered.
30## Question: Would you compare between example 1 and 2? path 2(0,1)1)2)(-1,0)(1,0)## Question: Would you compare between example 1 and 2?
312) Conservative fields (F or V) (보존장) - Example 1 : depends on the path. nonconservative field- Example 2 : does not depend on the path. conservative field
323) Potential () (퍼텐셜)for A: a proper reference pointcf. Electric field, gravitational field conservative
33Example 3. Show that F is conservative, and find a scalar potential.
34find the point where the field (or potential) is zero. 2) Scalar potential of F(0,0,0)(x,y,z)find the point where the field (or potential) is zero.do line integral to an arbitrary point along the path with which the integration is easiest.i) dxii) dyiii) dz(x,0,0)(x,y,0)i) only dxii) only dyiii) only dz
35Example 4. scalar potential for the electric field of a point charge q at the origin
369. Green’s theorem in the plane (평면에서의 Green 정리) 1) Definition of Green theorem- The integral of the derivative of a function is the function.
39This relation is valid even for an irregular shape!! “Using Green’s theorem we can evaluate either a line integral around a closed path or a double integral over the area inclosed, whichever is easier to do.”
40Example 1.F=xyi-y2j, find the work from (0,0) to (2,1) and backFor a closed path,(previous section)path 2 (parabola)path 3 (broken line)2)1)1)2)
41Example 1.F=xyi-y2j, find the work from (0,0) to (2,1) and backFor a closed path,Using Green’s theorem,
42Example 2.( z-component of curl F = 0),then, W from one point to another point is independent of the path.(F : conservative field)
43- Two useful way to apply Green’s theorem to the integration of vector functions a) Divergence theoremDivergence theorem
45Chapter 6 Vector analysis Mathematical methods in the physical sciences 3rd edition Mary L. BoasChapter 6 Vector analysisLecture 21 Divergence and Divergence theorem
4610. Divergence and divergence theorem (발산과 발산정리) 1) Physical meaning of divergenceflow of a gas, heat, electricity, or particles: flow of wateramount of water crossing A’ for t
47- Rate at which water flows across surface 1 - Net outflow along x-axisIn this way,“Divergence is the net rate of outflow per unit volume at a point.”
48cf. (from ‘Griffiths’)(a) positive divergence for positive charge (or negative divergence for a negative charge)(b) zero divergence(c) positive divergence along the z-axis
49cf. (from ‘Griffiths’)Example 1.4 in Figure 1.18
502) Example of the divergence 1 = (source density) minus (sink density)= net mass of fluid being created (or added via something like a minute sprinkler system) per unit time per unit volume = density of fluid = mass per unit volume/t = time rate of increase of mass per unit volumeRate of increase of mass in dxdydz= (rate of creation) minus (rate of outward flow)1) If there is no source or sinks,2)cf.
513) Example of the divergence 2 Consider any closed surface.Mass of fluid flowing out through d is Vn d.Total outflow:For volume element d=dxdydz, the outflow from d isanother definition of divergence
524) Divergence theorem5) Example of the divergence theoremIf we directly evaluate
536) Gauss’s lawelectric field at r due to a point chage q at (0,0)For multi-sources,
62Example 1.(a) integrate the expression at it stands(b) use Stoke’s theorem and evaluate the line integral around the circle(c) use Stoke’s theorem to say that the integral is the same over any surface bounded by the circle, for example, the planar area inside the circle.
63Ampere’s lawH : magnetic fieldC : closed curveI : current
64For a specific case,: one of the Maxwell equations
65Conservative fields‘simply connected’ if a simple closed curve in the region can be shrunk to a point without encountering any points not in the region.If the components of F have continuous first partial derivatives in a simple connected region, any one implies all the others.a) curl F = 0b) closed line integral = 0c) F conservatived) F = grad W, W single valued