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Mathematical methods in the physical sciences 3rd edition Mary L. Boas. Chapter 4 Partial differentiation ( 편미분 ). Lecture 12 Introduction of partial differentiation. 1. Introduction. - Differentiation. ex. - Time rates such as velocity, acceleration, and rate of cooling of a hot body.

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## Chapter 4 Partial differentiation ( 편미분 )

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**Mathematical methods in the physical sciences 3rd edition**Mary L. Boas Chapter 4 Partial differentiation (편미분) Lecture 12 Introduction of partial differentiation**1. Introduction**- Differentiation ex. - Time rates such as velocity, acceleration, and rate of cooling of a hot body. - Rate of change of volume of a gas with applied pressure (P, V) - Rate of decrease of fuel in your car tank with distance travelled (l, Q) - Differential equation - Finding Max. or Min.**- Partial differentiation**When we want to find the slope of z with respect to y, keeping x constant, we can use the partial differentiation. z y x**- High order partial derivatives**Example**The symbol is usually read “the**partial of z with respect to r, with x held constant”. However, the important point to understand is that the notation means that z has been written as a function of the variables r and x only, and then differentiated with respect to r. (Caution)**2. Power series in two variables (두 변에 대한**멱급수) Example 1. Example 2**- General expression**First, express the function with the power series and then determine the coefficients. Finding partial derivatives,**Then,**Using a simpler form, x – a = h and y – b = k. second-order terms, similarly, third-order terms Finally,**3. Total differentials (전 미분)**- Single variables - Two variables and more**4. Approximations using differentials (미소량을**이용한 어림) Example 1.**- Example 3. reduced mass**If m_1 is increased by 1%, what fractional change in m_2 leaves unchanged?**Example 4.**Relative error rate: 5 % in the length measurement and 10 % for the radius measurement**Mathematical methods in the physical sciences 3rd edition**Mary L. Boas Chapter 4 Partial differentiation Lecture 13 Chain rule**5. Chain rule or differentiating a function of a function**(연쇄법칙과 함수의 함수 미분하기) Example 1. ‘chain rule’**6. Implicit differentiation (음함수 미분)**Example 1. We realized that x is a function and just differentiate each term of the equation with respect to t (implicit differentiation). This problem is even easier if we want only the numerical values of the derivatives at a point.**7. More chain rule (더 많은 연쇄법칙)**Example 1.**Example 2.**- Using the differentials,**- Using the derivatives,**cf. Using the matrix form,**Example 3.**‘A computer may save us some time with the algebra.’**Let’s skip Example 5.**Example 6. Rectangular vs. polar coordinates. (reciprocal) constant y i) and ii)-1 are different!! constant r**This is a general rule: partial derivatives**are not usually reciprocals; they are reciprocals if the other independent variables (besides u or v) are the same in both cases.**Mathematical methods in the physical sciences 3rd edition**Mary L. Boas Chapter 4 Partial differentiation Lecture 14 Max. & Min. 26/15**8. Application of partial differentiation to maximum and**minimum problems (최대, 최소값 문제에서 편미분의 응용) - dy/dx=0 is a sufficient condition for max. or min. of f(x). inflection max. (convex) min. (concave) (d2y/dx2 = 0) (d2y/dx2> 0) (d2y/dx2< 0) - To minimize z = f(x,y), cf. saddle point 27/15**Example. A pup tent of given volume, V, with ends but**no floor, is to be made using the least possible material. find the proportions. To minimize A, 28/15**9. Maximum and minimum problems with constants; Lagrange**multipliers (제한조건이 있는 최대 최소값 문제 ; Lagrange 곱수) Example 1. shortest distance - Methods: (a) elimination, (b) implicit differentiation, (c) Lagrange multipliers (a) Elimination (제거방법) 29/15**Example 2. Shortest distance from the origin to the plane**cf. Equation of plane, ax+by+cz=d If (a,b,c) is a unit vector, abs(d) is a distance from the origin. 31/15**(c) Lagrange Multipliers**‘two functions’ ‘single function’ cf. valid for more than variables, ex. (x,y,z) 32/15**Example 3. Find the volume of the largest rectangular**parallelepiped (that is box) with edges parallel to the axes, inscribed in the ellipsoid, Multiplying each equation with the other variable, and then, adding all three, 34/15**- More constraints**35/15**To find the maximum or minimum of f subject to the**conditions Φ1=const. and Φ2=const., define F=f + λ1Φ1+ λ2Φ2and set each of the partial derivatives of F equal to zero. Solve these equation and the Φ equation for the variables and the λ’s. 36/15**Example 4. Minimized distance from the origin to the**intersection of 37/15**10. Endpoint or boundary point problems (끝점 혹은**경계점 문제) - Besides the extreme points, we should check the boundary points (or lines). case I case II case III case IV 38/15**Example 1. A piece of wire 40 cm long is to be used to**form the perimeters of a square and a circle in such a way as to make the total area (of a square and circle) a maximum. r (40-2r)/4 Considering the values at the boundary points, 39/15**Example 2. The temperature in a rectangular plate bounded**by the lines, (3,5) - Differentiating, y=5 x=3 - Boundary check (0,0) min. max. and check corners. At (3,5), T= 130. 40/15**H. W. (Due 5/21)**Chapter 4 4-15, 6-4, 9-11, 10-9 41/15**Mathematical methods in the physical sciences 3rd edition**Mary L. Boas Chapter 4 Partial differentiation Lecture 15 Change of variables**11. Change of variables (변수 변환)**Sometimes, we can make the differential equation simpler by changing variables. Example 1. Make the change of variables. Here, we can use the operation notation,**Then,**cf. compare with the original eq,**Example 2. Laplace equation**cf. Schrodinger eq.: cylindrical spherical**(ii)**For convenience,**1)**2) 1)**1)**2) 2) Finally,

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