Jacobian engineering mathematics

Описание: 📘 Jacobian | Partial Differentiation | VTU Model QP–I 2025

📝 Question Discussed in the Video
VTU Model Question Paper–I (2025 Scheme)
Subject Code: 1BMATS101 – Calculus & Linear Algebra
Module: 1
Question No.: 1(b)

If
u = x + 3y² − z³
v = 4x²yz
w = 2z² − xy

Find ∂(u, v, w) / ∂(x, y, z) at the point (1, −1, 0).

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📘 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  

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📘 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...  

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📘 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   

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📘 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...  

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🆕 NEW – VTU 1BMATS101 (2025 Scheme)

📘 Module 1 – Calculus
👉    • Calculus | VTU 1BMATS101 – Module 1 (Semes...  

📘 Module 3 – Linear Algebra
👉    • Linear Algebra Made Simple – Complete  

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📚 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...  

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