Hildebrand received his bachelor's degree in 1936 and a Master's degree in 1938 from Washington & Jefferson College, both in mathematics. He then received his Ph.D. degree from Massachusetts Institute of Technology in 1940 under the supervision of Prescott Durand Crout. He also received an honorary doctorate from Washington and Jefferson College in 1969.

At MIT, he taught 18.075 and 18.076, the classes on advanced calculus for engineering students. The big green textbook from these classes (originally Advanced Calculus for Engineers, later Advanced Calculus for Applications) was a fixture in engineer's offices for decades.

Hildebrand had authored many influential textbooks in mathematics, including

• Advanced Calculus for Engineers, Prentice Hall, 1948.
• Methods of Applied Mathematics, Prentice Hall, 1952.
• Advanced Calculus for Applications, Prentice Hall, 1964.
• Introduction to Numerical Analysis, 2 ed., Dover Publications, 1987 ISBN 0486653633 (First edition in 1956).