This paper delves into critical concepts and meticulous calculations pertinent to Simultaneous Localization and Mapping (SLAM), with a focus on error analysis and Jacobian matrices. We introduce various types of errors commonly encountered in SLAM, including reprojection error, photometric error, relative pose error, and line reprojection error, alongside their mathematical formulations. The fundamental role of error as the discrepancy between observed and predicted values in SLAM optimization is examined, emphasizing non-linear least squares methods for optimization. We provide a detailed analysis of: - Reprojection Error: Including Jacobian calculations for camera poses and map points, highlighting both theoretical underpinnings and practical consequences. - Photometric Error: Addressing errors from image intensity variations, essential for direct method-based SLAM. - Relative Pose Error: Discussing its significance in pose graph optimization, especially in loop closure scenarios. The paper also presents extensive derivations of Jacobian matrices for various SLAM components such as camera poses, map points, and motion parameters. We explore the application of Lie theory to optimize rotation representations and transformations, improving computational efficiency. Specific software implementations are referenced, offering practical insights into the real-world application of these theories in SLAM systems. Additionally, advanced topics such as line reprojection errors and IMU measurement errors are explored, discussing their impact on SLAM accuracy and performance. This comprehensive examination aims to enhance understanding and implementation of error analysis and Jacobian derivation in SLAM, contributing to more accurate and efficient state estimation in complex environments.