Chapter 6 Vector analysis (벡터 해석) Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 6 Vector analysis (벡터 해석) Lecture 18 Basic vector analysis
1. Introduction Vector function, Vector calculus, ex. Gauss’s law
2. Application of vector multiplication (벡터곱의 응용) 1) Dot product 2) Cross product - Example a) Work b) Torque c) Angular velocity
3. Triple products (삼중곱) 1) Triple scalar product (삼중 스칼라곱) “Volume of the parallelepipe” cf. volume of unit cell for reciprocal vectors
- An interchange of rows changes just the sign of a determinant. So, it does not matter where the dot and cross are.
2) Triple vector product (삼중 벡터곱) some vector in the plane of B and C Prove this! (Vector equation is true independently of the coordinate system.)
3) Application of the triple scalar product “Torque” This question is in one special case, namely when r and F are in a plane perpendicular to the axis.
Centripetal acceleration 4) Application of the triple vector product Angular momentum Centripetal acceleration
4. Differentiation of vectors (벡터의 미분) 1) Differentiation of a vector Example 1.
2) Differentiation of product
Example 2. Motion of a particle in a circle at constant speed Differentiating the above equations, “two vectors are perpendicular”
3) Other coordinates (e.g., polar) constant in magnitude and direction constant in magnitude, but directions changes
Example 3.
Chapter 6 Vector analysis Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 6 Vector analysis Lecture 19 Directional derivative; Gradient
5. Fields (장) Field: region + the value of physical quantity in the region ex) electric field, gravitational field, magnetic field
6. Directional derivative: gradient (방향 도함수 ; 기울기벡터) The change of temperature with distance depends on the direction. directional derivative 1) definition of directional derivative (directional derivative for u: directional unit vector)
Example 1. Find the directional derivative
2) Meaning of gradient : along it the change (slope) is fastest (steepest).
3) Relation between scalar function and gradient “The vector grad. is perpendicular to the surface =const.”
Example 3. surface x^3y^2z=12 Example 3. surface x^3y^2z=12. find the tangent plane and normal line at (1,-2,3)
4) other coordinates (e.g., polar) cf. Cylindrical & Spherical coord. cylindrical spherical
7. Some other expressions involving grad. ( 을 포함하는 다른 표현들) 1) vector operator 2) divergence of V 3) curl of V
4) Laplacian
5) and etc.
Chapter 6 Vector analysis Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 6 Vector analysis Lecture 20 Line integral & Green’s theorem
8. Line integrals (선적분) 1) definition integrating along a given curve. only one independent variable!
Example 1. F=(xy)i-(y2)j, find the work from (0,0) to (2,1) path 1 (straight line) path 2 (parabola)
path 3 (broken line) 1) 2) 2) 1) path 4 (parameter) x=2t^2, y=t^2 x: (0,2) t: (0,1)
Example 2. Find the value of path 1 (polar coordinate ) r=1 (constant) so, only d may be considered.
## 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?
2) Conservative fields (F or V) (보존장) - Example 1 : depends on the path. nonconservative field - Example 2 : does not depend on the path. conservative field
3) Potential () (퍼텐셜) for A: a proper reference point cf. Electric field, gravitational field conservative
Example 3. Show that F is conservative, and find a scalar potential.
find 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) dx ii) dy iii) dz (x,0,0) (x,y,0) i) only dx ii) only dy iii) only dz
Example 4. scalar potential for the electric field of a point charge q at the origin
9. 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.
Area integral: Line integral: cf.
Similarly,
This 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.”
Example 1. F=xyi-y2j, find the work from (0,0) to (2,1) and back For a closed path, (previous section) path 2 (parabola) path 3 (broken line) 2) 1) 1) 2)
Example 1. F=xyi-y2j, find the work from (0,0) to (2,1) and back For a closed path, Using Green’s theorem,
Example 2. ( z-component of curl F = 0), then, W from one point to another point is independent of the path. (F : conservative field)
- Two useful way to apply Green’s theorem to the integration of vector functions a) Divergence theorem Divergence theorem
b) Stoke’s theorem Stoke’s theorem
Chapter 6 Vector analysis Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 6 Vector analysis Lecture 21 Divergence and Divergence theorem
10. Divergence and divergence theorem (발산과 발산정리) 1) Physical meaning of divergence flow of a gas, heat, electricity, or particles : flow of water amount of water crossing A’ for t
- Rate at which water flows across surface 1 - Net outflow along x-axis In this way, “Divergence is the net rate of outflow per unit volume at a point.”
cf. (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
cf. (from ‘Griffiths’) Example 1.4 in Figure 1.18
2) 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 volume Rate of increase of mass in dxdydz = (rate of creation) minus (rate of outward flow) 1) If there is no source or sinks, 2) cf.
3) 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 is another definition of divergence
4) Divergence theorem 5) Example of the divergence theorem If we directly evaluate
6) Gauss’s law electric field at r due to a point chage q at (0,0) For multi-sources,
Gauss’s Law Using the divergence theorem,
7) Example of gauss’s law. E=? a) For electrostatic problem, E=0 inside b) For symmetry, E should be vertical. total charge inside is C(surface area) for C surface charge density.
Chapter 6 Vector analysis Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 6 Vector analysis Lecture 22 Curl and Stoke’s theorem
11. Curl and Stoke’s theorem (회전이론과 Stoke’s 정리) Ex. “Curl v gives the angular velocity.”
1) meaning of curl curl V 0 curl V = 0 curl V 0 curl V = 0 curl V 0
cf. (from ‘Griffiths’) vs.
cf. (from ‘Griffiths’) Example 1.5
bad good Stoke’s theorem
Example 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.
Ampere’s law H : magnetic field C : closed curve I : current
For a specific case, : one of the Maxwell equations
Conservative 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 = 0 b) closed line integral = 0 c) F conservative d) F = grad W, W single valued
Vector potential
Example 2. i) ii) There are many A’s to satisfy this equation. For convenience, set one component A_x =0.
There are many ways to select f1 and f2.
Generalization for A For A_x=0, cf. When we know one A, all others are of the form,
cf. Cylindrical coordinate
cf. Spherical coordinate
Homework Chapter 6 3-17, 4-5, 6-10, 10-14, 11-16