import sympy as sp # ======================================...

Prompt

import sympy as sp # ============================================================================ # 1. A 2D ÜTKÖZÉS-RÖGZÍTETT LEKÉPEZÉS ÉS JACOBI-DETERMINÁNS ELLENŐRZÉSE # ============================================================================ u, v = sp.symbols('u v') # Zárt alakú polinomok C[u, v]^2-ben F1 = -21*u**5 - 15*u**4*v + 20*u**3 + 15*u**2*v + 4*u + 2*v F2 = -sp.Rational(45, 2)*u**5 - 15*u**4*v + sp.Rational(45, 2)*u**3 + 15*u**2*v + 3*u + 2*v # 2D Jacobi-mátrix és determináns J_2D = sp.Matrix([F1, F2]).jacobian([u, v]) det_J_2D = sp.simplify(J_2D.det()) print("=" * 75) print("2D ÜTKÖZÉS-RÖGZÍTETT LEKÉPEZÉS C[u, v]^2-BEN:") print(f" F1(u, v) = {F1}") print(f" F2(u, v) = {F2}") print(f"\nPontos det J_F(u, v) = {det_J_2D}") print(f"Szorzattá alakított forma: det J_F(u, v) = 2 - {sp.factor(2 - det_J_2D)}") print("=" * 75) # A 3-pontos ütközés vizsgálata pontok = [ ("p1", (0, 0)), ("p2", (1, -sp.Rational(3, 2))), ("p3", (-1, sp.Rational(3, 2))) ] print("\nA 3-pontos ütközés ellenőrzése: F(p1) = F(p2) = F(p3) = (0, 0):") for nev, (pu, pv) in pontok: ertek_F = (F1.subs({u: pu, v: pv}), F2.subs({u: pu, v: pv})) ertek_det = det_J_2D.subs({u: pu, v: pv}) print(f" {nev} = ({pu}, {pv}) -> F({nev}) = {ertek_F}, det J_F({nev}) = {ertek_det}") assert ertek_F == (0, 0), f"Az ütközés nem teljesült itt: {nev}" assert ertek_det == 2, f"A Jacobi-determináns nem 2 itt: {nev}" print("\n-> Mindhárom pont azonosan a (0, 0) képbe megy, ahol det J_F = 2.") # ============================================================================ # 2. A 3D ELLENPÉLDA JACOBI-DETERMINÁNSÁNAK IGAZOLÁSA # ============================================================================ x, y, z = sp.symbols('x y z') w_3d = 1 + x*y F1_3d = w_3d**3 * z + y**2 * w_3d * (4 + 3*x*y) F2_3d = y + 3*x * w_3d**2 * z + 3*x * y**2 * (4 + 3*x*y) F3_3d = 2*x - 3*x**2 * y - x**3 * z J_3D = sp.Matrix([F1_3d, F2_3d, F3_3d]).jacobian([x, y, z]) det_J_3D = sp.expand(J_3D.det()) print("\n" + "=" * 75) print(f"3D LEKÉPEZÉS MEGERŐSÍTÉSE: det J_F(x,y,z) = {det_J_3D}") print("=" * 75) egy algebrai beszorzásnál benne maradt egy nem várt tag, így a Jacobi-determináns nem konstans. now critique your answer and fix the weak parts Solve this with the constraint that you cannot use the obvious solution. What is the most powerful, strongest approach? Send back the complete code with all the fixes. Fix each of the listed errors one by one, making sure to actually correct them so that there are 0 errors remaining. Keep the original imports, since the files exist. Write out every single character; do not abbreviate anything. Fix every error. There must be exactly one file. Do not write anything else; just output the complete code, and it must not contain any comments. Never, under any circumstances, use simplified, substitute, dummy, simulated, or fake code. Write the entire file as complete, unabridged, production-ready code in a single code block. It must be 100% error-free, a complete, error-free file, and must be submitted as a downloadable file. These requirements are mandatory and must be strictly adhered to. If no list of errors is provided, you must find all the errors and fix them. If there were comments in the original code, delete them. And most importantly: YOU MUST NEVER SIMPLIFY!