http://matrix.skku.ac.kr/Mobile-Sage-G/sage-grapher.html

 Define Function Taylor expansion   Equation solving Approximate solution Partial fraction   Limit Right hand limit Left hand limit Limit at + infinity Limit at - infinity   Derivative Second derivative Indefinite integral Definite integral   Series   Radius of convergence       Define a vector Norm of the vector Distance from yz-plane Distance from zx-plane Distance from xz-plane Distance from x-axis Distance from y-axis Distance from z-axis   Define a vector valued function Velocity  Acceleration  Speed Velocity vector at t = 2 Acceleration vector at t = 2 Arc length Dot product Cross product f(x)=exp(-2*x) f(x).taylor(x, 2, 4)   solve(f(x)==0, x) find_root(f(x), a, b) f.partial_fraction(x)   limit(f(x), x=2) limit(f(x), x=2, dir='+') limit(f(x), x=2, dir='-') limit(f(x), x=+oo) limit(f(x), x=-oo)   diff(f(x), x) diff(f(x), x, 2) integral(f(x), x) integral(f(x), (x, -2, 3))   sum((e/10)^x, x, 1, +oo)   u(x)=1/(x*2^x) rho=limit(abs(u(x+1)/u(x)), x=+oo) 1/rho   x = vector([3, -4, 5]) x.norm() sqrt(x[0]^2) sqrt(x[1]^2) sqrt(x[2]^2) sqrt(x[1]^2 + x[2]^2) sqrt(x[0]^2 + x[2]^2) sqrt(x[0]^2 + x[1]^2)   var('t'); r=vector([2-5*t,4*t,1+3*t]) v=diff(r, t) a=diff(r, t, 2) v.norm() v.subs(t=2) a.subs(t=2) integral(v.norm(), (t, 0, 5)) v.dot_product(a)  v.cross_product(a)

 Find gradient     Double integral Triple integral   Double integral (in Polar coordinate)       Cylindrical  to Rectangular   Spherical  to Rectangular   Graph in plane Parametric function in plane Implicit function in plane Line segment in plane   Graph in Parametric function in Implicit function in   Level curve   Vector field Vector field in   Line integral       curl (A) div (A)   Green’s theorem Stokes’ theorem Divergence theorem var('x, y, z')     f=x*y+y*z^2+x*z^3 f.gradient() integral(integral(f, (x, 0, y)), (y, 0, 1)) integral(integral(integral(f, (x, 0, y)), (y, 0, 1)), (z, 0, 1))   var('r, t') f=arctan(tan(t)) integral(integral(f*r, (r, 1, 2)), (t, 0, pi/4))   T = Cylindrical('height', ['radius', 'azimuth']) T.transform(radius=1, azimuth= pi, height=2)   T = Spherical('radius', ['azimuth', 'inclination']) T.transform(radius=3, azimuth=pi/6, inclination=pi/6)   plot(sin(x), (x, -4, 4)) parametric_plot((sin(x), cos(x)), (x, 0, 2*pi)) var('x, y'); implicit_plot(sin(x)-y==1, (x, -2, 2), (y, -2, 2)) line([(1, 1), (2, 2)], color='red')   var('x, y'); plot(x^2+y^2, (x, -2, 2), (y, -2, 2)) parametric_plot3d((sin(x), cos(x), x), (x, 0, 2*pi)) var('x, y, z'); implicit_plot3d(x^2+y^2==5, (x, -3, 3), (y, -3, 3),  (z, -3, 3))   var('x, y'); contour_plot(x^2+y^2, (x, -1, 1), (y, -1, 1), cmap='hsv', labels=True)   var('x, y'); plot_vector_field((x+y, x), (x, -3, 3), (y, -3, 3))   var('x, y, z'); plot_vector_field3d((0, 0, 1), (x, -3, 3), (y, -3, 3), (z, -3, 3))   var('t'); r=vector([t-t^3, t^2, 0]) integral(r.dot_product(diff(r, t)), (t, 0, pi))   A(x,y,z) = P*i+Q*j+R*k  # conservative if curl(F)=0. curlA=(diff(R,y)-diff(Q,z))*i+(diff(P,z)-diff(R,x))*j+(diff(Q,x)-diff(P,y))*k  divA = diff(P,x)+diff(Q,y)+diff(R,z)    # divergence of A   integral(integral((diff(N, x)-diff(M, y))*r, (r, 0, 3)), (t, 0, 2*pi)) integral(integral(curl(F).dot_product(-n), (r, 0, 1)), (t, 0, 2*pi)) integral(integral(integral(Div, 0, 3), (y, 0, 2)), (z, 0, 1))

 Rules for Inequalities   1. If , then . 2. If , then  . 3. If , then . 4.  If , then . 5. If , then . 6. If , then . 7. If , then . 8. If , then  . 9. If , then .     Special Functions   1. Exponential Functions If and , then a function of the form is called an exponential function. The number is called the base and is called the exponential. http://matrix.skku.ac.kr/Mobile-Sage-G/sage-grapher.html   2. Logarithmic Functions The logarithmic with base and , is defined by .   3. Inverse Trigonometric Functions                   4. Hyperbolic Functions 5. Inverse Hyperbolic Functions                                                                Formulas of Trigonometric Functions   1. Addition and Subtraction Formulas   2. Double Angle Formulas   3. Half Angle Formulas   4. Triple Angle Formulas   5. Product-to-Sum

 6. Sun-to-Product                     7. Linear Combination      where,      where,     Limits http://matrix.skku.ac.kr/cal-lab/SKKU-Cell-Epsilon-Delta.html http://matrix.skku.ac.kr/cal-lab/cal-Limit.html   1. Limit Laws Suppose that the limits and exist and is a constant. Then    if , where is a positive integer. , where is a positive integer.     (If is even, we require ) http://matrix.skku.ac.kr/cal-lab/cal-2-1-7.html http://matrix.skku.ac.kr/cal-lab/cal-2-2-3.html   2. Squeeze Theorem (or Sandwich Theorem) If when is near and 　 then http://matrix.skku.ac.kr/cal-lab/cal-2-1-12.html Derivatives 1. Differentiation Rules   where is a positive integer and is a constant.   2. Chain Rule  If is any real number and is differentiable and , then .   3. Parametric Formula For the parametric equations: and ,  ,   4. Implicit Function   Let be an implicit function.       where .   5. Inverse Function      or    6. Trigonometric Functions                                     http://matrix.skku.ac.kr/cal-lab/cal-3-3-Exm24.html

 7. Inverse Trigonometric Functions                        http://matrix.skku.ac.kr/cal-lab/cal-3-2-13.html   8. Logarithmic Functions                                9. Exponential Functions                               10. Hyperbolic Functions                                11. Inverse Hyperbolic Functions                                                           General Rules of Integration http://matrix.skku.ac.kr/cal-lab/cal-RiemannSum.html http://matrix.skku.ac.kr/cal-lab/cal-6-3-17.html 1. Basic Forms    http://matrix.skku.ac.kr/cal-lab/cal-7-7-Exm-8.html , http://matrix.skku.ac.kr/cal-lab/cal-5-2-20.html   2. Trigonometric Forms http://matrix.skku.ac.kr/cal-lab/cal-5-4-exm-7.html

 3. Hyperbolic Forms http://matrix.skku.ac.kr/cal-lab/cal-8-1-9.html   4. Forms Involving 5. Inverse Trigonometric Forms http://myhandbook.info/form_integ.html     Series   1. Taylor Series http://matrix.skku.ac.kr/cal-lab/cal-10-5-Exm-11.html   2. Maclaurin Series ,                  ,                  ,  ,      ,       3. Binomial Series If  and are any real numbers and is a positive integer, we have              where and     Vectors   1. Dot Product Let and . http://matrix.skku.ac.kr/cal-lab/cal-11-3-2.html

 2. Projections Scalar projection of onto : , Vector projection of onto : http://matrix.skku.ac.kr/cal-lab/cal-11-5-20.html   3. Definition and Properties of Cross Product http://matrix.skku.ac.kr/cal-lab/cal-11-4-Exs-6.html Let two nonzero vectors and are two sides of a parallelogram, then the area of the parallelogram is . http://matrix.skku.ac.kr/cal-lab/cal-11-4-10.html   4. Rules of Limits , http://matrix.skku.ac.kr/cal-lab/9-5-Example-7.html   5. Rules of Differentiation http://myhandbook.info/form_diff.html  where is a scalar http://matrix.skku.ac.kr/cal-lab/cal-4-3-24.html http://matrix.skku.ac.kr/cal-lab/cal-4-2-9.html   6. Derivative of a Vector Function If , then http://matrix.skku.ac.kr/cal-lab/cal-13-2-Exm6.html http://matrix.skku.ac.kr/cal-lab/cal-12-4-3.html http://matrix.skku.ac.kr/cal-lab/cal-13-2-20.html 7. Integral of a Vector Function If ,   then   8. Arc Length http://matrix.skku.ac.kr/cal-lab/cal-8-1-9.html http://matrix.skku.ac.kr/cal-lab/cal-13-3-2.html   9. Curvature , http://matrix.skku.ac.kr/cal-lab/cal-13-3-12.html   10. Equations of Line  : a vector equation               : parametric equations      : a symmetric equation http://matrix.skku.ac.kr/cal-lab/11-5-Exmaple-14.html   11. Equation of Plane  : a standard form of a plane                 : a vector version of a plane    : parametric equations of a plane http://matrix.skku.ac.kr/cal-lab/cal-11-5-20.html     Formulas of Vector Calculus   1. Line Integral http://matrix.skku.ac.kr/cal-lab/Sec15-2-1.html http://matrix.skku.ac.kr/cal-lab/Sec15-4-Exs-1.html

 2. Path Independent Theorem Let be a potential function for . For any piecewise smooth curve from and ,       3. Area of a Plane Region If has a piecewise smooth boundary with positive orientation, then the area of is       4. Area of a Surface http://matrix.skku.ac.kr/cal-lab/cal-8-2-Exm-4.html http://matrix.skku.ac.kr/cal-lab/cal-0-a.html    5. Surface Integral http://matrix.skku.ac.kr/cal-lab/cal-14-5-1.html http://matrix.skku.ac.kr/cal-lab/Sec15-6-Exm-4.html http://matrix.skku.ac.kr/cal-lab/Sec15-6-Exm-5.html http://matrix.skku.ac.kr/cal-lab/Sec15-6-Exm-8.html http://matrix.skku.ac.kr/cal-lab/Sec15-7-Exm-2.html http://matrix.skku.ac.kr/cal-lab/cal-15-9-Exam-3.html   6. Gradient http://matrix.skku.ac.kr/cal-lab/cal-12-2-2.html   7. Divergence http://matrix.skku.ac.kr/cal-lab/Sec15-5-Exm-3.html   8. Curl http://matrix.skku.ac.kr/cal-lab/Sec15-8-Exm-2.html 9. Laplace Operator http://matrix.skku.ac.kr/cal-lab/cal-12-2-6.html   10. Vector Triple Products http://matrix.skku.ac.kr/cal-lab/cal-11-4-16.html     Theorems on Vector Calculus   1. Green's Theorem   http://matrix.skku.ac.kr/cal-lab/Sec15-4-Exm-1.html http://matrix.skku.ac.kr/cal-lab/Sec15-4-Exs-10.html http://matrix.skku.ac.kr/cal-lab/cal-15-2-10.html   2. Stoke's Theorem   http://matrix.skku.ac.kr/cal-lab/Sec15-5-Exm-3.html http://matrix.skku.ac.kr/cal-lab/cal-14-7-4.html   3. Divergence Theorem   div .    Mobile Sage Grapher :http://matrix.skku.ac.kr/Mobile-Sage-G/sage-grapher.html  http://math1.skku.ac.kr/home/pub/1433/