MITx Online | Certificate for: Multivariable Calculus 3: Theorems and Applications

Massachusetts Institute of Technology

This is to certify that Takayuki Mashima has successfully completed Multivariable Calculus 3: Theorems and Applications Issued: Oct. 29, 2025

Lawrence Guth

C. E. Shannon (1940) Professor in Mathematics

Massachusetts Institute of Technology

Dimitris Bertsimas

Vice Provost for Open Learning

Associate Dean, Online Education and Artificial Intelligence

Boeing Professor of Operations Research

Massachusetts Institute of Technology

Valid Certificate ID: 74d36d6f-5989-48d2-824f-e90cf8db3600