The chapter "Singularities of Differentiable Maps" by J. M. Boardman explores the local structure of smooth maps between manifolds, focusing on the concept of singularity subsets. For each point \( p \in V \), the differential \( f_p \) maps the tangent space \( T_p \) to \( T_{f(p)} \). The rank of \( f_p \) determines the singularity subset \( \Sigma^i(f) \), which consists of points where the kernel of \( f_p \) has dimension \( i \). Thom proved that for most maps \( f \), these sets \( \Sigma^i(f) \) are submanifolds of \( V \). Boardman introduces a more technical approach using jet spaces, which are constructed through iterated Jacobian extensions. These jet spaces induce subsets \( \Sigma^i(f) \) of \( V \) for any map \( f \) and finite sequence of integers. The main theorem asserts that these subsets are submanifolds in the jet space, and their codimensions are computed. This implies that \( \Sigma^i(f) \) are submanifolds of \( V \) for "sufficiently good" maps \( f \). The chapter also discusses the inductive procedure by Porteous for constructing these sets and intrinsic derivatives, which works for "good" maps. The results are valid algebraically and in other contexts like complex or real cases. The notation and terminology used are defined, including the use of infinite jet spaces and vector bundles.The chapter "Singularities of Differentiable Maps" by J. M. Boardman explores the local structure of smooth maps between manifolds, focusing on the concept of singularity subsets. For each point \( p \in V \), the differential \( f_p \) maps the tangent space \( T_p \) to \( T_{f(p)} \). The rank of \( f_p \) determines the singularity subset \( \Sigma^i(f) \), which consists of points where the kernel of \( f_p \) has dimension \( i \). Thom proved that for most maps \( f \), these sets \( \Sigma^i(f) \) are submanifolds of \( V \). Boardman introduces a more technical approach using jet spaces, which are constructed through iterated Jacobian extensions. These jet spaces induce subsets \( \Sigma^i(f) \) of \( V \) for any map \( f \) and finite sequence of integers. The main theorem asserts that these subsets are submanifolds in the jet space, and their codimensions are computed. This implies that \( \Sigma^i(f) \) are submanifolds of \( V \) for "sufficiently good" maps \( f \). The chapter also discusses the inductive procedure by Porteous for constructing these sets and intrinsic derivatives, which works for "good" maps. The results are valid algebraically and in other contexts like complex or real cases. The notation and terminology used are defined, including the use of infinite jet spaces and vector bundles.