Here is a list of 100 command prompts for performing mathematical analysis, categorized by its major branches. These prompts are framed as tasks or instructions for a mathematician, data scientist, or computational software.
1. Real Analysis: Foundations
Prove that the set [S] is countable or uncountable.Find the supremum, infimum, min, and max of the set [S].Prove the [Completeness/Archimedean] property for the real numbers.Analyze the convergence of the sequence [a_n = formula].Find the limit of the sequence [a_n] as n approaches infinity.Prove that the sequence [a_n] is a Cauchy sequence.Determine if the series [Σ a_n] converges or diverges using the [Ratio/Root/Comparison] Test.Test the series [Σ a_n] for absolute and conditional convergence.Find the radius and interval of convergence for the power series [Σ c_n * (x-a)^n].Find the limit of the function [f(x)] as x approaches [a] using an (ε, δ) proof.Prove that the function [f(x)] is continuous at the point [x=a].Determine if the function [f(x)] is uniformly continuous on the interval [I].Apply the Intermediate Value Theorem to show a root exists for [f(x)] on [a, b].Apply the Extreme Value Theorem to find the global extrema of [f(x)] on [a, b].
2. Differential Calculus
Find the first and second derivatives of the function [f(x)].Calculate the derivative of [f(x)] from the limit definition.Find the equation of the tangent line and normal line to [y=f(x)] at [x=a].Apply the [Product/Quotient/Chain] Rule to differentiate [function].Perform implicit differentiation for the equation [F(x, y) = 0].Find all critical points of the function [f(x)].Determine the intervals where [f(x)] is increasing or decreasing.Classify the local extrema of [f(x)] using the [First/Second] Derivative Test.Analyze the concavity and find all inflection points of [f(x)].Apply L'Hôpital's Rule to evaluate the limit of [f(x)/g(x)].Find the [Nth] degree Taylor polynomial for [f(x)] centered at [x=a].Calculate the error (remainder) for the Taylor approximation of [f(x)].
3. Integral Calculus
Calculate the indefinite integral of [f(x)] dx.Evaluate the definite integral of [f(x)] from [a] to [b].Apply the Fundamental Theorem of Calculus to evaluate [integral].Use the method of integration by parts to evaluate [integral].Use trigonometric substitution to evaluate [integral].Use partial fraction decomposition to evaluate [integral].Determine if the improper integral [integral] converges or diverges, and evaluate if possible.Calculate the area of the region bounded by [y=f(x)] and [y=g(x)].Calculate the volume of the solid generated by rotating the region [R] around the [x-axis/y-axis].Calculate the arc length of the curve [y=f(x)] from [a] to [b].
4. Multivariable & Vector Calculus
Calculate the partial derivatives (∂f/∂x, ∂f/∂y) for the function [f(x, y)].Find the gradient vector (∇f) for the function [f(x, y, z)].Find the directional derivative of [f(x, y)] at point [P] in the direction of vector [v].Find the equation of the tangent plane to the surface [z=f(x, y)] at the point [P].Find and classify all critical points (local min/max, saddle) of the function [f(x, y)].Use the method of Lagrange multipliers to find the extrema of [f(x, y)] subject to the constraint [g(x, y) = c].Evaluate the double integral of [f(x, y)] over the rectangular region [R].Change the order of integration for the double integral [integral].Convert the double integral to polar coordinates and evaluate.Evaluate the triple integral of [f(x, y, z)] over the volume [V].Convert the triple integral to [cylindrical/spherical] coordinates and evaluate.Calculate the Jacobian of the transformation [T(u, v) = (x(u, v), y(u, v))].Evaluate the line integral of [f(x, y)] along the curve [C] parameterized by [r(t)].Calculate the curl and divergence of the vector field [F].Determine if the vector field [F] is conservative.Find the potential function for the conservative vector field [F].Use Green's Theorem to evaluate the line integral [integral] over the closed curve [C].Use Stokes' Theorem to evaluate the curl integral over the surface [S].Use the Divergence Theorem to evaluate the flux integral over the closed surface [S].
5. Complex Analysis
Express the complex number [z] in [polar/Euler's] form.Verify the Cauchy-Riemann equations for the function [f(z)] at [z_0].Determine all points where the function [f(z)] is analytic.Evaluate the complex line integral of [f(z)] over the contour [C].Apply Cauchy's Integral Theorem to evaluate the integral of [f(z)] over [C].Apply Cauchy's Integral Formula to evaluate the integral of [f(z)/(z-z_0)^n] over [C].Find the Taylor series expansion for [f(z)] centered at [z_0].Find the Laurent series expansion for [f(z)] in the annulus [R1 < |z-z_0| < R2].Identify and classify all singularities (removable, pole, essential) of the function [f(z)].Calculate the residues of [f(z)] at all its isolated singularities.Use the Residue Theorem to evaluate the contour integral of [f(z)].
6. Differential Equations (ODEs & PDEs)
Classify the differential equation (order, linearity, homogeneity).Solve the separable first-order ODE [y' = f(x)g(y)].Solve the linear first-order ODE [y' + p(x)y = q(x)] using an integrating factor.Find the general solution to the homogeneous second-order linear ODE [ay'' + by' + cy = 0].Find the particular solution to the non-homogeneous ODE using [undetermined coefficients/variation of parameters].Solve the initial value problem (IVP) for the given ODE and initial conditions.Find the equilibrium solutions for the autonomous ODE [y' = f(y)] and analyze their stability.Draw the phase portrait for the 2D linear system [x' = Ax].Use the method of separation of variables to solve the [Heat/Wave/Laplace's] equation.Solve the [PDE] subject to the given [Dirichlet/Neumann] boundary conditions.
7. Fourier Analysis & Transforms
Find the Fourier series representation of the periodic function [f(x)] on the interval [-L, L].Analyze the convergence of the Fourier series and describe the Gibbs phenomenon.Find the complex Fourier series for the function [f(x)].Calculate the Fourier transform of the function [f(t)].Calculate the inverse Fourier transform of [F(ω)].Apply the convolution theorem to find the transform of [f(t) * g(t)].Calculate the Laplace transform of the function [f(t)].Use the Laplace transform to solve the ODE [IVP].
8. Functional Analysis & Linear Algebra
Prove that the set [V] with the given operations forms a vector space.Find the basis and dimension of the vector space [V] spanned by [vectors].Prove that [||.||] is a norm on the vector space [V].Prove that [d(x, y)] is a metric on the set [M].Determine if the metric space [M, d] is complete (i.e., a Banach space).Prove that [<.,.>] defines an inner product on the space [H].Determine if the inner product space [H] is a Hilbert space.Apply the Gram-Schmidt process to find an orthonormal basis for [subspace].Find the norm of the linear operator [T: V -> W].Find the kernel (null space) and image (range) of the linear operator [T].Find all eigenvalues and eigenvectors (or eigenspaces) of the operator [T].
9. Numerical Analysis
Apply the Bisection method to find a root of [f(x)=0] on the interval [a, b].Apply Newton's method to find a root of [f(x)=0] starting at [x_0].Find the Lagrange interpolating polynomial for the data points [points].Use the [Trapezoidal/Simpson's] rule to numerically approximate the integral of [f(x)] from [a] to [b].Use [Euler's method/Runge-Kutta method] to numerically solve the ODE [y' = f(x, y)] with initial condition [y(x_0) = y_0].
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