diff --git a/北京师范大学/偏微分方程/Chapter_1.tm b/北京师范大学/偏微分方程/Chapter_1.tm new file mode 100644 index 0000000..56625b3 --- /dev/null +++ b/北京师范大学/偏微分方程/Chapter_1.tm @@ -0,0 +1,181 @@ + + + + +<\body> + + |>>> + + + + Suppose we deposite > in a bank account, and the annual + interest rate is . After years, + + <\enumerate> + If the interest is compounded annualy, then the balance is + + <\equation*> + u=u*. + + + If the interest is compounded monthly, then the balance is + + <\equation*> + u=u*|)> + + + In general, if the interest is compounded times a year, + then the balance is + + <\equation*> + u=u*|)>. + + + Taking the limit as \>, then for any + fixed , we have + + <\equation*> + u=lim\>|)>=lim\>|)>>>*r*t>=e. + + + + But we can obtain the same result by using a differential equation: + + <\equation> + =u*r>. + + + We can solve it later and obtain the same result. + + <\definition> + A (ODE) about a function + > is an equation involving + > and its derivatives. + + + + + Any ODE can be written in the abstract form + + <\equation> + F,u,\,u>|)>=0. + + + For example, Eq. (1) can be written as + + <\equation*> + F|)>=r*u-u=0. + + + In Eq. (2), is the of the equation, i.e. the + order is the highest order derivative of in the equation.\ + + So Eq. (1) is an ODE of order 1.\ + + If is linear in terms of ,\,u>>, + then the equation is called . Otherwise it's called + . The general form of a linear ODE is + + <\equation*> + a*u>+a*u>+\+a*u+a=0. + + + So Eq. (1) is linear. Examples of nonlinear equations: + + <\equation*> + u-u=5,u*u+5x=e,u=sin + u + + + + + A solution of an ODE + + <\equation*> + F,u,\,u>|)>=0 + + + is a function > satisfying the equation, + i.e. + + <\equation*> + F,\,\,\,\>|)>=0. + + + <\example> + Can you give solutions of + + <\equation*> + u=2*u + + + possible solution: >, in fact, + > is a solution for any constant . + + + <\example> + Can you give solutions of + + <\equation*> + u+4*u=0 + + + A possible solution is , another solution is + =cos*2t.> In fact, any function + + <\equation*> + u=a*sin 2*t+b*cos 2*t + + + + \; + + \; + + \; + + \; + + \; + + \; + + \; + + \; + > + + +<\initial> + <\collection> + + + + + +<\references> + <\collection> + > + > + > + + + +<\auxiliary> + <\collection> + <\associate|toc> + |1Motivation + |.>>>>|> + > + + |2Classification of ODE + |.>>>>|> + > + + |3Solutions of an ODE + |.>>>>|> + > + + + \ No newline at end of file diff --git a/北京师范大学/微积分/Chapter_1.tm b/北京师范大学/微积分/Chapter_1.tm new file mode 100644 index 0000000..81570fd --- /dev/null +++ b/北京师范大学/微积分/Chapter_1.tm @@ -0,0 +1,392 @@ + + +> + +<\body> + + |>>> + + + + <\definition> + A function is a to assign to a variable, say , an + value, say . In general, we can write + + <\equation*> + y=f. + + + Here + + <\itemize> + is the name of the function + + is called the + + is called the + + + The set of values that can take is called the + of the function. The set of all possible values of constitute + the . + + > is read + + + We can view a function as a machine, that takes the input , and + gives the output . + + The of a + > is the curve + + <\equation*> + \\:y=f|}>. + + + >: the plane + + + + |gr-frame|>|gr-geometry||gr-color|red|||>>|||>>||||>>||>>|>>|>>||>>|>>|>||>>||>>|>>| + This represents the\ + + graph of a function + >>||>||>||||>>||>>|>|>|>|>>||>>||>>||>>| + This curve doesn't represent the + + graph of a function + >>>> + + <\em> + + + + <\equation*> + y=|0>>||0>>>>> + + + This function can also be written as + + <\equation*> + y=. + + + (even/odd functions) + + functions + + + + <\itemize> + linear functions + + <\equation*> + y=m*x+b + + + where is the , and is the + -. + + For a linear function, if =m*x+b,y=m*x+b>, + then + + <\equation*> + y-y=m*-x|)>. + + + Graphing calculator: www.desmos.com + + power functions + + <\equation*> + y=x + + + The property depends on the value of . + + <\itemize> + If >, then it's an even function, and + it's increasing function for 0> and decreasing for + 0>. + + If >, then it's an odd function, and + it's increasing function for all . + + If >, then it's an even function, and + it's decreasing function for 0> and increasing for + 0>. + + If >, then it's an odd function, and + it's decreasing function for 0> or 0>. + + If ,,,\>, then + the function is defined only for 0>, and it's an + increasing function. + + If ,,,\>, then + the function is defined for all , and it's an increasing + function. + + other cases: read the slides or book + + + polynomials + + <\equation*> + y=a*x+a*x+\+a*x+a, + + + where >is called the of the + polynomial, and >'s are called the + . For example, -x+5> is a + polynomial of degree 2, or we call it a , + and +2x-10> is a polynomial of degree 3, also + called a . + + The domain of any polynomial is =,\|)>>. + + rational functions + + <\equation*> + y=|Q>, + + + where are both polynomials. For example + + <\equation*> + y=-2x|4x-5> + + + The domain of a rational function is all except where + =0>. + + algebraic functions: any function the can be obtained by + addition, subtraction, multiplication, and raising to rational powers + of . For example + + <\equation*> + y=+3*|x-4>+10*x. + + + exponential functions + + <\equation*> + y=b, + + + where 0> is called the , is called + the . + + logarithmic functions + + <\equation*> + y=log*x + + + they are actually the of exponential + functions, i.e. + + <\equation*> + y=log x\x=b + + + trigonometric functions + + <\equation*> + y=sin x,cos x,tan x,cot x,sec x,csc x + + + + Elementary building blocks:\ + + <\enumerate> + power functions + + exponential functions + + logarithmic functions + + trigonometric functions + + + + + <\itemize> + translations (move the graph up/down, left/right) + + <\itemize> + vertical translation + + <\equation*> + y=f+c + + + horizontal translation + + <\equation*> + y=f + + + + stretching and reflecting + + <\itemize> + vertical stretching/reflecting + + <\equation*> + y=c*f + + + horizontal stretching/reflecting + + <\equation*> + y=f|)> + + + + algebraic combinations + + <\equation*> + f+g,f-g,f*g, + + + composition + + <\equation*> + f|)>=g> + + + read as of of , or composed with + + + <\example> + If =x+3*x> and + =sin x>. Then + + <\itemize> + g=f>|)>=>|)>+3*>=+3*sin x> + + f=g|)>=sin + f=sin +3*x|)>.> + + f=|)>+3*f=+3*x|)>+3*+3*x|)>> + + g=sin g=sin + .> + + + <\equation*> + x|g>g|gr-arrow-end|\||||>>|g|>>>|2cm>|f>f|)> + + + + + + + + <\example> + Write the function =log + +1>> as a composition of 3 functions. + + Answer: Let =log + x,g=,h=x+1>. Then + + <\equation*> + F=f\g\h + + + + Application example (compound interest): Suppose you deposit an + amount of > in an account with annual interest rate of + . The balance after years, if the interest is + compounded + + <\itemize> + annualy, then + + <\equation*> + u=u* + + + monthly, then + + <\equation*> + u=u*|)>=u*|)>|]> + + + times per year, then + + <\equation*> + u=u|)> + + + continuously, then + + <\equation*> + u=lim\>u|)>=lim\>u|)>|]>=u*e, + + + where > is called the . + + + Here we are using the formula + + <\equation*> + \>|)>=e> + + + + \; + + \; + + \; + + \; + + \; + + \; + + \; + > + + +<\initial> + <\collection> + + + + + + + + + +<\references> + <\collection> + > + > + > + + + +<\auxiliary> + <\collection> + <\associate|toc> + |math-font-series||1Functions + and Models> |.>>>>|> + + + |math-font-series||2A + catalog of essential functions> |.>>>>|> + + + |math-font-series||3New + functions from old> |.>>>>|> + + + + \ No newline at end of file