f := x->;
fsolve(D(f)(x) = 0, x = range);
(D@@2)(f)(代入);

f := ;
s := fsolve(f = 0, z, complex);
s0 := map(abs, s);
max(s0);

with(plots);
plot(f, x= -1..1);
//记得要算两边端点

r := ;
polarplot(r, t = 0..2*Pi);

f := ;
g := ;
implicitplot({f,g}, x = -1..2, y = -1..2);

plot([x(t), y(t), t = 0..2 * Pi]);
maximize(f(x), x = 0..2*Pi, location);
minimize(f(x), x = 0..2*Pi, location);
solve({f(x) = 0, x >= 1, x <= 10}, x);

image: Pasted image 20251126034332.png

Pi, I, infinity, exp();
a := 1; //命名一个变量

f := x ^ 2 + y;
subs(x = 3, f); // 显示 9 + y;
subs(x = 3, y =1, f); //显示10;

evalf(Pi, 100); //精确到100位有效数字

simplify(f);
expand(f);
factor(f);

f := x-> x ^ 2;
f(5);
g :=(x, y)-> x * sin(y);
g(Pi, 1);

limit(f, x = 0);
limit(f, x = 0, left);
limit(f, x = -infinity);

//对表达式求导
diff(f, x);
diff(f, x$2);
implicitdiff(sin(x*y)=1, y, x);

//对函数求导
D(f);
D(f)(5);//代入5
(D@@2)(f);
int(f, x); //不定积分
int(f, x = 0..1); //定积分

set := {1,2,3};
list := [1,2,3];
sort(list); //只有list能排序

sum(i, i = 1..100); // 1 + 2 + ... + 100;
product(i, i = 1..100) // 1 * 2 * 3 * ... * 100;

eqn1 := x + y + z = 1;
eqn2 := 2 * x + y + z = 4;
eqn3 := 3 * x + y + z = 7;
solve({eqn1, eqn2, eqn3}, {x, y, z});
fslove({eqn1, eqn2, eqn3}, {x, y, z});

fsolve(x^5 - x - 1 = 0, x); // 寻找一个实数根
fsolve(x^2 + 1 = 0, x, complex); // 寻找复数根: -I, I
fslove(x^ 2  = 0, x, -1..10); // 限制解的范围

z := 3 + 4 * I;
Re(z);
Im(z);
abs(z);
argument(z);
conjugate(z);

with(plots);
plot(sin(x), x = -Pi..Pi);
plot(tan(x), x = -Pi..Pi, y = -10..10); // 限制绘图范围

//两图联立
f := x;
g := 2 * x;
plot({f,g},x = -100..100);

//参数方程（用[ ]把方程和变量括在一起）
plot([cos(t), sin(t), t = -Pi..Pi]);

ploarplot(1 + cos(t), t = 0..2 * Pi); // t 是角度，1 + cos(t)是距离

implicitplot(x ^ 2 + y ^ 2 = 1, x = -1..1, y = -1..1);

Matrix
<<1,2,3> | <4,5,6> | <7,8,9>> // 列表示法
<<1|2|3>, <4|5|6>, <7|8|9|>> // 行表示法


for i from 0 to n by step do
a[i] = a[i - 1] + 1:
end do:

f := proc(x)
  local i, s;
  s := 0;
  for i from 1 to x do
      s := s + i:
  end do:
  return s;
end proc;

Digits := 30;
f := proc(m)
  local a, i;
  a[0] := 0;
  for i from 1 to m do
    a[i] := sin((1 + (1/3)*a[i-1])^2):
  end do:

  if abs(a[m]-a[m-1]) < 10^(-19)
    return a[m]
  else
    return -1
  end if

end proc;

Calculus 笔记

可分离 ODE / 一阶 ODE

ode := diff(y(x), x) = x^4 * y(x)^2;
ic := y(3) = 5;
dsolve({ode, ic}, y(x));

两种写法都行：D(y)(x) 或 diff(y(x), x)

二阶常系数非齐次 ODE

ode := (D@@2)(y)(x) + 2*D(y)(x) + y(x) = 8*exp(-x);
dsolve(ode, y(x));

如果有初始条件就加上：dsolve({ode, y(0)=..., D(y)(0)=...}, y(x))

非线性二阶 ODE

ode := y(x) * (D@@2)(y)(x) + (D(y)(x))^2 = 0;
ic_1 := y(0) = 2;
ic_2 := D(y)(0) = 3;
dsolve({ode, ic_1, ic_2}, y(x));

答案只填等号右边，不含 y(x) =

Taylor 级数系数

taylor(exp(4*x)*sin(5*x), x = 0, 7);
coeftayl(exp(4*x)*sin(5*x), x = 0, 6);

taylor 第三个参数是阶数，要看到 x⁶ 就写 7
coeftayl 直接取某一项的系数

递推数列

a[1] := 4: a[2] := -1: a[3] := 2:
for i from 3 to 99 do
  a[i+1] := a[i] - 2*a[i-1] + a[i-2]
end do:
a[100];

循环里必须用 :=，用 = 不会赋值，只是一个等式表达式。

Exact ODE

M := 4*x^3*y^2 + 4;
N := 2*x^4*y - 4;
simplify(diff(M, y) - diff(N, x));
# 结果为 0 -> 恰当

矩阵与向量空间

Rank(A) -> rank
ColumnDimension(A) - Rank(A) -> nullity
NullSpace(A) -> ker(A) 的基
ColumnSpace(A) -> im(A) 的基

判断向量是否在 ker(A) 中：

算 NullSpace(A) 看该向量能否由零空间基线性表示，或直接算 A . Vector([...]) 看是否为零

判断向量是否在 im(A) 中：

LinearSolve(A, b) 有解 -> 在 im(A) 中，报错 -> 不在

特征值，特征向量： Eigenvalues Eigenvectors：左边输出 Eigenvalues，右边输出对应的特征向量

MAPLE 1