Course coordinator

Heather Bolles

Catalog description

Geometry of space and vectors, multivariable differential calculus, multivariable integral calculus, vector calculus.
Prerequisite: Minimum of C- in MATH 166 or high math placement scores
Credit hours: 4

Course description

The tools of calculus, including differentiation and integration, are applied to functions of several variables and vector-valued functions.


The etext version of Pearson’s textbook Thomas’ Calculus, Early Transcendentals, 14th Edition, by George Thomas, Jr., Maurice D. Weir and Joel Hass accompanies a purchased access code. The access code provides access to the online homework package. Students should access the online homework (MyMathLab) only through Canvas.


Geometry of space; vectors (Sections 12.1 – 12.6, 13.1 – 13.5, 14.1, 14.2, 15.7) includes discussion of different coordinate systems of space, and measuring distance. Discussion of vectors includes operations with vectors and applications of vectors. Lengths, velocity, and acceleration are calculated using parameterized curves.

Multivariable differentiation (Sections 14.3 – 14.9, 15.1 – 15.3) includes discussion of partial derivatives, the chain rule with several variables, and approximating multivariable functions using Taylor polynomials. Differentiation is applied to identify extreme values.

Multivariable integration (Sections 15.4 – 15.8, 16.1 – 16.8) includes discussion of setting up and evaluating double and triple integrals to solve various application-oriented problems.

Learning objectives

After completing Math 265, students should be able to:

  • Calculate the distance between two points in Cartesian coordinates or between a point and a plane in Cartesian form.
  • Describe a set of points in space whose coordinates satisfy a given equation or inequality or a combination of inequalities or equations.
  • Convert an equation in Cartesian coordinates to an equivalent equation in cylindrical and spherical coordinates. Convert an equation in cylindrical or spherical coordinates to an equivalent equation in Cartesian coordinates.
  • Calculate the volume of the solid described by a combination of equations or inequalities.
  • Perform operations with vectors: compute the length and direction of a vector, add two or more vectors, multiply with a scalar, calculate the dot product and the cross product of two vectors.
  • Describe the geometric interpretation of the dot product and the cross product and use the understanding to solve practical problems.
  • Given two vectors, compute the angle between the two vectors and the projection of one vector onto the other.
  • Use operations on vectors to solve problems about work or torque.
  • Given a triangle or a parallelogram, compute the area.
  • Given two points, write the parametric and vector equation of the line through these two points.
  • Given three points, write the equation of the plane through these three points.
  • Calculate the distance from a point to a line or a point to a plane.
  • Describe a surface in space from its equation.
  • Describe the graph of a vector function from its equation.
  • Given a particle’s position as a vector function, compute its velocity, speed, and tangent line at a given time.
  • Solve differential equations with initial value problems related to the vector position of a particle.
  • Given two pieces of information (initial position, initial velocity, position at a given time, or launch angle and speed), determine the projectile motion of an object at any time t, and answer follow-up questions about its motion.
  • Given a space curve, calculate its length.
  • Given a space curve, calculate its speed and unit tangent vector at a given time.
  • Given the vector position r at a specific time, give the osculating plane.
  • Given the vector position r, calculate the tangential and normal component of acceleration.
  • Given a parameterized curve, calculate the curvature, κ.
  • Sketch the domain, graph and level sets (curves) of a function of several variables.
  • Compute the partial derivatives of a function using the appropriate rules of differentiation.
  • Apply the chain rule for functions of several variables properly.
  • Find the tangent plane to f(x,y) at (a,b) where f_x and f_y are continuous near (a,b) for a multivariable function z=f(x,y).
  • Calculate the directional derivative of a function in a given direction, the gradient of a function, the linearization of a function and the Taylor expansion of a function.
  • Compute the tangent plane and normal directions to a surface at a given point.
  • Use the gradient and the linearization to estimate change and solve problems.
  • Use the Taylor expansion to approximate functions at a point.
  • Apply the first derivative test to find critical points of a function; apply the second derivative test to determine whether the critical point is a minimum, maximum, saddle point, or if the test is inconclusive.
  • Find all local and global extrema of a function within a given region or curve.
  • Apply the method of Lagrange multipliers to solve constrained extrema problems (this involves but is not restricted to solving systems of equations and justifying that the critical point found is indeed the desired min/max).
  • Calculate double integrals over rectangular regions.
  • Choose an appropriate set of coordinates and set up the double integral of a function over a given region.
  • Calculate double integrals over rectangular regions.
  • Given a region in the plane, choose an appropriate set of coordinates and set up the double integral of a function over that region.
  • Given a double integration, sketch the region over which the integration happens and reverse the order of integration.
  • Use double integrals to calculate area and solve applications.
  • Calculate triple integrals.
  • Given a region in Euclidean space, set up an appropriate triple integral of a function over that region in rectangular coordinates.
  • Use triple integrals to calculate volume, center of mass, moments of inertia and solve applications for regions in 3-D.
  • Given a double integration, sketch the region over which the integration happens and reverse the order of integration.
  • Given a three-dimensional region, choose an appropriate set of coordinates and set up the triple integral of a function over that region.
  • Compute the Jacobian of a change of variables.
  • Perform a change of variables (substitution) in a double or triple integral.
  • Compute the gradient vector field of a function, the curl and divergence of vector fields.
  • Set up and evaluate line an integral of a function f  defined parametrically over a curve C .
  • Calculate mass and moments for coil springs, wires, and thin rods lying along a smooth curve in space.
  • For a continuous force field F  over a region containing a smooth curve C  parameterized by r(t)  compute the work done in moving an object from one point on the curve to another point on the curve.
  • Determine whether a vector field is conservative. If so, compute its potential.
  • Apply Green’s Theorem to compute the flux across or the circulation of a vector field along a closed curve by transforming into a double integral over a region in a plane.
  • Compute the surface area of parametrized surfaces and graphs of functions.
  • Set-up and compute the integral of a function over a surface. The surface can be defined parametrically, given implicitly, or given explicitly as a graph.
  • Apply Stokes’ Theorem to calculate the circulation of a vector field or to find the integral of the curl of the vector field over a surface.
  • Apply the Divergence Theorem to compute the flux of a vector field across the boundary of a region or the divergence of the vector field in the region.


Three midterm night exams and a cumulative final exam are given. Times for the night exams may be found at the night exam schedule at

Weekly quizzes are administered in recitation sessions.

Three online (MyMathLab) homework assignments are due most weeks. The online homework package with a purchased access code includes the electronic textbook.

Many materials including videos, practice exams, and quizzes are available at

Math 265 Help

Math 265 students may access the help hour times and location by following the link established in the Canvas course pages.

Free expression statement

Iowa State University supports and upholds the First Amendment protection of freedom of speech and the principle of academic freedom in order to foster a learning environment where open inquiry and the vigorous debate of a diversity of ideas are encouraged. Students will not be penalized for the content or viewpoints of their speech as long as student expression in a class context is germane to the subject matter of the class and conveyed in an appropriate manner.

Students with disabilities

Iowa State University is committed to assuring that all educational activities are free from discrimination and harassment based on disability status. Students requesting accommodations for a documented disability are required to work directly with staff in Student Accessibility Services (SAS) to establish eligibility and learn about related processes before accommodations will be identified. After eligibility is established, SAS staff will create and issue a Notification Letter for each course listing approved reasonable accommodations. This document will be made available to the student and instructor either electronically or in hard-copy every semester. Students and instructors are encouraged to review contents of the Notification Letters as early in the semester as possible to identify a specific, timely plan to deliver/receive the indicated accommodations. Reasonable accommodations are not retroactive in nature and are not intended to be an unfair advantage. Additional information or assistance is available online at, by contacting SAS staff by email at, or by calling 515-294-7220. Student Accessibility Services is a unit in the Dean of Students Office located at 1076 Student Services Building.

More information about disability resources in the Mathematics Department can be found at