A Jacobian Matrix Approach to Transform Vector Calculus Operators from Cartesian to Spherical Coordinates

Marcu Kaoklai Sima — 2026-09-02

This paper will explore the Jacobian matrix from both geometric and analytical perspective, focusing on how it arises from the transformation of infinitesimal volume elements. Then, as a practical application, we will demonstrate how the Jacobian can be used to transform key vector calculus operators from Cartesian to spherical coordinates.

Analytical Thinking in Mathematics: Learning Activities on Finding the General Term of S-Pattern Numbers

Mathasit Tanyarattanasrisakul — 2026-09-02

Analytical thinking is a critical skill for learning mathematics and can be fostered through appropriately designed learning activities. This article aims to present concepts and approaches for organizing learning activities on finding the general term of S-pattern numbers that promote analytical thinking skills. The key content includes the concept of analytical thinking, refers to an individual’s ability to dissect information into smaller components to understand principles or relationships and to rationally expand knowledge, leading to the creation of new information or conclusions. In this article, the authors adopt the framework of Marzano and Kendall, which categorizes analytical thinking skills into five aspects: matching, classifying, analyzing errors, generalizing, and specifying. These aspects are used to design learning activities involving teacher questioning and group tasks related to S-shaped figurate numbers, numbers represent a specific type of arrangement of square tiles on a plane forming the shape of the letter “S” in English.

Mathematical Journal by The Mathematical Association of Thailand Under The Patronage of His Majesty The King

A Jacobian Matrix Approach to Transform Vector Calculus Operators from Cartesian to Spherical Coordinates

Analytical Thinking in Mathematics: Learning Activities on Finding the General Term of S-Pattern Numbers