Here is a sample document for mathematics notes for GATE (Graduate Aptitude Test in Engineering) specifically designed for mechanical engineers:

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GATE Mathematics Notes for Mechanical Engineering

Topic: Linear Algebra

1. Matrices and Determinants
- Introduction to matrices and determinants
- Types of matrices: square, symmetric, skew-symmetric, etc.
- Operations on matrices: addition, subtraction, multiplication
- Properties of determinants: expansion, properties of minors and cofactors
- Inverse of a matrix: methods to find inverse, properties of inverse
- Rank of a matrix: methods to find rank, properties of rank

2. Systems of Linear Equations
- Introduction to systems of linear equations
- Methods to solve linear equations: Gauss elimination, Gauss-Jordan elimination, Cramer's rule
- Consistency and inconsistency of systems of equations
- Rank of a system of equations
- Homogeneous and non-homogeneous systems of equations

3. Eigenvalues and Eigenvectors
- Introduction to eigenvalues and eigenvectors
- Eigenvalues and eigenvectors of a matrix
- Diagonalization of a matrix
- Properties of eigenvalues and eigenvectors
- Application of eigenvalues and eigenvectors in engineering

4. Vector Spaces
- Introduction to vector spaces
- Subspaces and span of vectors
- Linear independence and linear dependence
- Basis and dimension of a vector space
- Orthogonal and orthonormal vectors
- Gram-Schmidt orthogonalization process

5. Inner Product Spaces
- Introduction to inner product spaces
- Inner product and its properties
- Norm and distance in inner product spaces
- Orthogonal and orthonormal bases
- Gram-Schmidt process in inner product spaces
- Application of inner product spaces in engineering

Topic: Calculus

1. Differential Calculus
- Limits and continuity
- Differentiation of functions: rules and techniques
- Maxima and minima of functions
- Application of differentiation in engineering

2. Integral Calculus
- Definite and indefinite integrals
- Techniques of integration: substitution, integration by parts, partial fractions, etc.
- Improper integrals
- Application of integration in engineering

3. Differential Equations
- Introduction to differential equations
- First-order linear and nonlinear differential equations
- Second-order linear homogeneous and non-homogeneous differential equations
- Application of differential equations in engineering

4. Multivariable Calculus
- Partial derivatives and their applications
- Gradient, divergence, and curl of a vector field
- Line, surface, and volume integrals
- Green's theorem, Stokes' theorem, and divergence theorem
- Application of multivariable calculus in engineering

These are just a few topics covered in the mathematics section of GATE for mechanical engineering. It is advisable to refer to standard textbooks and previous years' question papers to get a comprehensive understanding of the subject.

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Please note that this is just a sample document and the actual notes may vary depending on the specific requirements of the GATE examination and the individual preferences of the student. It is recommended to refer to reliable study materials and consult with subject experts for a more detailed and tailored approach to GATE preparation.