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Traffic flow linear algebra

📘 Traffic Flow Problem Using System of Linear Equations | Linear Algebra | VTU 1BMATS101 📝 Question Discussed in the Video VTU Model Question Paper–I (2025 Scheme) Subject Code: 1BMATS101 – Calculus & Linear Algebra Module: 3 – System of Linear Equations Question No.: 5(c) 📌 Given Flow Equations x₁ + x₄ = 1000 x₁ + x₂ = 1600 x₂ + x₃ = 2200 x₃ + x₄ = 1600 (**incoming traffic = outgoing traffic** at each junction.) ━━━━━━━━━━━━━━━━━━━━━━ 📘 Module 1 | Calculus | VTU 1BMATS101 (2025 Scheme) 1️⃣ Partial Differentiation | Q 1(a) Question: Prove that ∂u/∂x + ∂u/∂y = u, when u = e^(x+y) / [(e^x)(e^y)]. 👉    • Partial differentiation engineering mathem...   2️⃣ Jacobian | Q 1(b) Question: If u = x + 3y² − z³, v = 4x²yz, w = 2z² − xy, find ∂(u,v,w)/∂(x,y,z) at (1, −1, 0). 👉    • Jacobian engineering mathematics   3️⃣ Maxima and Minima | Q 1(c) Question: Find the extreme values of f(x,y) = xy + a³/x + a³/y. 👉    • Maxima and minima engineering mathematics    4️⃣ Chain Rule | Q 2(a) Question: If v = f(r,s,t), r = x/y, s = y/z, t = z/x, show that x(∂v/∂x)+y(∂v/∂y)+z(∂v/∂z)=0. 👉    • Chain rule of partial differentiation   5️⃣ Jacobian | Q 2(b) Question: If u = 2yz/x, v = 3zx/y, w = 4xy/z, find ∂(u,v,w)/∂(x,y,z). 👉    • Jacobian   6️⃣ Maclaurin Series | Q 2(c) Question: Expand cos x · cos y using Maclaurin’s series up to second-degree terms. 👉    • maclaurin series of two variables   ━━━━━━━━━━━━━━━━━━━━━━ 📘 Module 2 | Vector Calculus | VTU 1BMATS101 (2025 Scheme) 1️⃣ Gradient of Dot Product | Q 3(a) Question: If f = x²yz and g = xy − 3z², calculate ∇(∇f • ∇g). 👉    • Vector calculus | Gradient of Dot Product   2️⃣ Irrotational Field & Scalar Potential | Q 3(b) Question: Given f = (6xy + z³)i + (3x² − z)j + (3xz² − y)k, show that the field is irrotational and find its scalar potential. 👉    • Vector calculus engineering mathematics | ...   3️⃣ Cartesian to Spherical Coordinates | Q 3(c) Question: Express the vector F⃗ = x î + 2y ĵ + yz k̂ in spherical polar coordinates. 👉    • cartesian to spherical coordinates | vecto...   4️⃣ Directional Derivative | Q 4(a) Question: Find the directional derivative of f(x,y,z) = 4e^(2x − y + z) at (1,1,−1) in the direction towards (−3,5,6). 👉    • Vector calculus engineering mathematics | ...   5️⃣ Divergence & Curl | Q 4(b) Question: Find the divergence and curl of vector f, where f = grad(x³ + y³ + z³ − 3xyz). 👉    • Vector calculus engineering mathematics | ...   6️⃣ Cartesian to Cylindrical Coordinates | Q 4(c) Question: Express the vector F⃗ = 2x î + 3y ĵ − z k̂ in cylindrical polar coordinates. 👉    • Cartesian to cylindrical coordinates | Vec...   ━━━━━━━━━━━━━━━━━━━━━━ 📘 Module 3 | System of Linear Equations, Eigenvalues and Eigenvectors | VTU 1BMATS101 (2025 Scheme) Part 1 👉 Rank of Matrix | Q 5(a) Find constant b if rank of a 4×4 matrix is 3    • Rank of matrix | Linear algebra    Part 2 👉 Diagonalization of Matrix | Q 5(b) Find modal matrix and verify diagonalization    • Diagonalization of Matrices | Linear Algebra   Part 3 👉 Traffic Flow Problem | Q 5(c) Form and solve system of linear equations    • Traffic flow linear algebra   Part 4 👉 Test for Consistency | Q 6(a) Investigate values of λ and μ    • Test for consistency and solve the equatio...   Part 5 👉 Gauss–Jordan Method | Q 6(b) Solve system using Gauss–Jordan method    • Gauss Jordan Method   Part 6 👉 Eigenvalues & Eigenvectors | Q 6(c) Find eigenvalues and eigenvectors    • Eigen values & eigen vectors    ━━━━━━━━━━━━━━━━━━━━━━ 📘 Module 4 | Vector Space | VTU 1BMATS101 (2025 Scheme) Part 1 👉 Linear Combination of Vectors (Q.7a)    • Linear combination of the vector | Vector ...   Part 2 👉 Subspace of R3 (Q.7b)    • Subspace of r3 | Vector space   Part 3 👉 Basis & Dimension of Row, Column and Null Space (Q.7c)    • Basis dimension row space column space nul...   Part 4 👉 Basis & Dimension of a Vector Space (Q.8a)    • Basis and dimension of a vector space | Ve...   Part 5 👉 Inner Product in Vector Space (Q.8b)    • Inner product in vector space | Vector space   Part 6 👉 Coordinates of a Vector w.r.t. a Basis (Q.8c)    • Vector Space | Coordinates of a vector wit...   ━━━━━━━━━━━━━━━━━━━━━━ 🆕 NEW – VTU 1BMATS101 (2025 Scheme) 📘 Module 1 – Calculus 👉    • Calculus | VTU 1BMATS101 – Module 1 (Semes...   📘 Module 3 – Linear Algebra 👉    • Linear Algebra Made Simple – Complete   ━━━━━━━━━━━━━━━━━━━━━━ 📚 All 1BMATS101 Playlists 👉 Model QP–I (2025):    • 1BMATS101 | VTU Calculus & Linear Algebra ...   👉 Module 1:    • Partial Differentiation | Multivariable Ca...   👉 Module 2:    • Vector Calculus | VTU 1BMATS101 – Module 2...   👉 Module 3:    • Linear Algebra: System of Equations & Eige...   👉 Module 4:    • Vector Space | VTU 1BMATS101 – Module 4 (S...   👉 Module 5:    • Linear Transformation | VTU 1BMATS101 – Mo...   ━━━━━━━━━━━━━━━━━━━━━━ 💎 Support Us Join our channel for exclusive perks 👇 🔗    / @officialmathematicstutor   #TrafficFlow #SystemOfEquations #1BMATS101 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