MTH164: Multidimensional Calculus

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Cross Listed:
Offered:
Fall and Spring
Prerequisites:
This course is a prerequisite or co-requisite for:
Two years of the calculus sequence MTH161--MTH164 or MTH141-143 and MTH163--MTH164 or MTH171--174 is required to major in mathematics. EE212, EE231, EE223, ME225, ME226, and ME280.
Description:
This extends the calculus techniques to handle functions of more than one variable. It also concentrates increasingly on the geometric aspect of calculus, the ability to picture what the symbols stand for. This ability to picture the information contained in the equations is particularly important for applying calculus to problems in physics, engineering (e.g. hydrodynamics), computer graphics and in upper level mathematics subjects such as differential geometry (MTH255).
Topics covered:
Usually MTH164 (multidimensional calculus) is taken before MTH163 (differential equations) since its subject matter is more closely related to MTH162. However some engineering majors require MTH163 to be completed by the end of the fall semester of the sophomore year. Differentiation and linear approximation, extrema, Taylor series. Line, surface, and volume integrals; coordinate changes, Jacobians. Divergence theorem, Stokes' theorem. Determinants and matrices in n-dimensional vector spaces.
Related courses:
  • The subjects discussed in MTH164 are further studied in MTH255 (differential geometry) and MTH266 (advanced analysis).
  • MTH164 includes a partial introduction to the subject of vector spaces; this subject is covered more completely in MTH235. It is possible to take these courses concurrently. The approach used in MTH235 (linear algebra) is more abstract, and special cases of the material covered in MTH235 have applications in MTH164 and MTH163 as well almost all advanced mathematics courses.
  • PHY122 or PHY142 (Electricity and magnetism) uses many of the techniques from this course. In particular line and surface integrals are used to calculate the total charges on surfaces; the divergence theorem is the basis for Gauss's Law, while Stokes' theorem is the basis for Ampere's Law. Taking this course concurrently with MTH164 gives immediate applications for many of the techniques taught in the math course. Frequently, you'll learn to use the techniques in the physics course and find out why they work in the math course.
  • These concepts are also heavily used in fluid dynamics.