graph LR
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    idea_0cd68a6b_2f4d_4942_9287_e28f51dc948b["Vector Arithmetic in Three-Dimensional Space"]
    idea_15588046_daa5_4b3b_b572_989a4ac3ed1f["Matrix Dimensions and Element Notation in Linear Algebra"]
    idea_1d3fccba_cec3_4106_96ef_240f17d77aa9["Multivariable Chain Rule for Composite Functions"]
    idea_2133580a_00a5_438b_8536_1df36a7f0cb2["Vector Equation of a Plane and a Line in Three-Dimensional Space"]
    idea_2183d640_ebc6_4d97_a0d3_ae9831a2e56c["Differentiating Integrals with Respect to a Parameter in Multivariable Calculus"]
    idea_331a052f_085c_43bd_9f0f_45d4a5d1bd4d["Fundamental Theorem for Iterated Double Integrals in Multivariable Calculus"]
    idea_461077ef_b371_4778_8c93_83d4bec44c2c["Matrix Operations in Linear Algebra"]
    idea_5484008d_05f0_491b_b128_bc2f86b047d5["Inverting Nonlinear Systems of Equations Using the Jacobian Matrix in Multivariable Calculus"]
    idea_5bcf23a8_7d65_45c4_b8ef_c0c4fb141719["Separation of Variables in Partial Differential Equations"]
    idea_623474f2_ceff_4a63_8ffe_52cd29d9f499["Optimality conditions for free problems in two or more independent variables"]
    idea_6cf7a862_819a_4159_88a1_355285f37dd5["Dot Product Operations in Euclidean Space"]
    idea_76f1b6a1_c304_43bc_96a3_3cf18653e226["Exact Differentials in Multivariable Calculus"]
    idea_870ad61b_3db7_465d_8366_a5acca0d8a8f["The Jacobian Determinant for Change of Variables in Multiple Integration"]
    idea_8f4bd6fa_5acb_4eec_a93f_57fef89012c3["Integral as Accumulated Summation Limit"]
    idea_93291d36_49ce_472d_bd30_5cb84b11680e["Vector Arithmetic in Multivariable Calculus"]
    idea_a560728e_2c2a_4f86_9b7e_cbbb90bb0e51["Velocity and Acceleration Vectors in Polar Coordinates for Multivariable Calculus"]
    idea_ab2996c4_d004_4e94_8721_916a2cb2e46e["N-Dimensional Vector Spaces and Nuples in Multivariable Calculus"]
    idea_b252bb59_a868_4b21_b1b6_dd8c4648fece["Partial Derivatives Multivariable Analysis"]
    idea_bd6f564f_eea9_48b5_9840_adbd964dfe2e["Computing the Inverse of a Matrix by Row Reduction in Linear Algebra"]
    idea_be9b0785_069a_4387_9bf6_057a61b6f95c["Cross Product in Three Dimensional Space"]
    idea_ce668018_f858_4f9b_8c92_cd8be22c05a9["Calculus of vector functions"]
    idea_d2b8da48_cd85_409e_a033_88e5339173b0["Vectors as Geometric Arrows and Ordered Number Lists"]
    idea_d3041c66_d79a_4777_a651_e913a1a60b47["Chain Rule Application for Composite Functions in Calculus"]
    idea_d3887949_a73d_45b2_9e94_34b39077b43a["Local Linear Approximation and Matrix Algebra in Multivariable Calculus"]
    idea_e2f529ec_3d46_41b1_8bcc_183d15c564a5["Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)"]
    idea_e61062eb_f0a0_439a_b6f2_040d86605109["Cartesian Coordinate Systems in Multivariable Calculus"]
    idea_f17f3f8b_e3a2_45c3_bdc6_fdf6479691f2["Derivatives and Partial Derivatives of Scalar Functions"]
    idea_f6642d69_2c53_4b4d_90e7_9235a20dd633["Polar Coordinates and Their Relation to Cartesian Coordinates in Multivariable Calculus"]
    idea_15588046_daa5_4b3b_b572_989a4ac3ed1f --> idea_461077ef_b371_4778_8c93_83d4bec44c2c
    idea_5bcf23a8_7d65_45c4_b8ef_c0c4fb141719 --> idea_b252bb59_a868_4b21_b1b6_dd8c4648fece
    idea_d2b8da48_cd85_409e_a033_88e5339173b0 --> idea_be9b0785_069a_4387_9bf6_057a61b6f95c
    idea_e61062eb_f0a0_439a_b6f2_040d86605109 --> idea_b252bb59_a868_4b21_b1b6_dd8c4648fece
    idea_d3041c66_d79a_4777_a651_e913a1a60b47 --> idea_1d3fccba_cec3_4106_96ef_240f17d77aa9
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