Iowa State University (ISU) Math Tutor & Mastery Lab | Calculus II: MATH 1660/1660H; Calculus III & Vector Calculus: MATH 2650/2650H; Differential Equations: MATH 2660/2670/3850; Linear Algebra: MATH 2070/3170/4070/5070; Abstract Algebra: MATH 3010/4030/5030/5040/5050; Real Analysis & Advanced Calculus: MATH 4140/4150/5010/5150/5160; Galois Theory: graduate MATH 5050; Number Theory: MATH 3500 (COMS 3500)

MATH 1660: Calculus II is Iowa State’s main Calculus 2 course. MATH 1660H: Calculus II: Honors adds further depth. The department’s standard course page describes night midterms and a cumulative final, so make review a weekly habit rather than a last-minute event. Follow your own section’s dates and instructions.

Practice a mixed set of integrals and explain why you chose substitution, integration by parts, a trigonometric method, or partial fractions. For volumes by washers and cylindrical shells, draw the region and axis, label the distances, and decide which slice makes the setup clearer. Work and fluid-force problems need that same attention to what each factor means.

For a series, distinguish a test for convergence from a formula for its sum. Check the endpoints of a power-series interval separately and explain what a Taylor approximation leaves out. In the Mastery Lab, study a complete example, rebuild it without looking, then try a nearby problem with one important feature changed.

MATH 2650: Calculus III and MATH 2650H: Calculus III: Honors are Iowa State’s multivariable calculus routes. The standard course’s published objectives connect spatial geometry to partial derivatives, multiple integrals, and vector-field calculations.

Start by asking what changes and what stays fixed. For a partial derivative, state which variable is held constant. For a directional derivative, make the direction a unit vector before calculating. For an optimization problem, separate the function you are optimizing from the constraint and check the candidates against the region.

When setting up a double or triple integral, sketch a slice and describe its endpoints in words. Compare Cartesian, cylindrical, and spherical coordinates before choosing bounds. In vector calculus, distinguish work along a curve from flux through a surface; draw the direction of travel or the normal. Work with Woody on explaining when Green’s, Stokes’s, or the divergence theorem applies, then check that your geometry satisfies its conditions.

Iowa State offers MATH 2660: Elementary Differential Equations and MATH 2670: Elementary Differential Equations and Laplace Transforms. MATH 2670 adds transform and power-series methods. MATH 2680: Laplace Transforms and MATH 2690: Systems of Differential Equations are focused one-credit supplements, rather than complete alternative differential-equations sequences.

Before calculating, classify the equation and write down the conditions. For a first-order problem, explain why separation or an integrating factor fits. For a linear system, connect its eigenvalues to the shape of the solution. Substitute your answer back into both the equation and the conditions to catch errors early.

MATH 3850: Introduction to Partial Differential Equations moves into PDE work. Identify the domain and boundary conditions before separating variables; then explain how the conditions select the terms in your expansion. Bring the exact topic and syllabus so your guidance matches the level you need.

MATH 2070: Matrices and Linear Algebra emphasizes techniques and applications, while MATH 3170: Theory of Linear Algebra places greater emphasis on proof. They are distinct Iowa State routes, so bring the course number along with the problem.

After row reduction, ask what you now know about consistency, free variables, and the dimension of the solution set. When deciding whether vectors form a basis, check both independence and spanning. Work with Woody on translating those calculations into statements about kernels, images, and linear maps.

MATH 4070/5070: Applied Linear Algebra is a dual-listed undergraduate/graduate course. For advanced matrix work, distinguish an eigenvalue decomposition from a singular value decomposition, and state the hypotheses before applying a canonical-form result. The Lab gives you a place to ask about the step you cannot justify and practice an explanation you can reproduce independently.

Start with MATH 3010: Abstract Algebra I. Continue with MATH 4030/5030: Intermediate Abstract Algebra, an undergraduate/graduate dual listing. The graduate sequence is MATH 5040: Abstract Algebra I and MATH 5050: Abstract Algebra II. The repeated title does not make MATH 3010 and MATH 5040 equivalent courses.

In Iowa State’s introductory algebra work, familiar integers and polynomials can give you concrete examples of more abstract structures. Practice explaining why a map preserves the operation, how its kernel organizes a quotient, or what an irreducibility argument actually establishes. For a proposed proof, first identify the definition that turns the conclusion into something you can check.

As your work progresses to intermediate or graduate algebra, use counterexamples to keep the hypotheses honest. Does an argument require commutativity, a multiplicative identity, or an inverse? Where does a claim about elements become a claim about an entire structure? Bring the precise theorem and your attempted proof into the Mastery Lab so Woody can help you develop the reasoning and reconstruct it independently.

MATH 4140: Analysis I and MATH 4150: Analysis II are Iowa State’s undergraduate analysis sequence. Graduate references include MATH 5010: Introduction to Real Analysis, MATH 5150: Real Analysis I, and MATH 5160: Real Analysis II.

For an epsilon proof, write down what has already been fixed, what you are allowed to choose, and what the final inequality must accomplish. Work backward from that inequality to find a useful estimate, then present the proof in forward order. A short example can help you find the right bound; it cannot replace the general argument.

In MATH 4150 and later work, pay attention to which notion of convergence a problem requires. When you want to interchange a limit with another operation, identify the theorem and verify its hypotheses first. Graduate measure and integration problems need the same discipline at a deeper level. Bring your current definition, notation, and attempted proof so the Lab guidance meets you at the correct stage.

MATH 5050: Abstract Algebra II is the graduate course reference here. An Iowa State faculty syllabus explicitly identifies Galois theory in this algebra sequence; the Spring 2025 syllabus confirms the current MATH 5050 number, formerly MATH 505. Bring your current syllabus because the depth and examples can vary. MATH 4030/5030 provides related field-extension foundations.

Begin with the base field. Find the minimal polynomial of the element you adjoin, establish irreducibility when needed, and explain the resulting extension degree. When several roots are involved, check which ones the field already contains before adjoining more.

For an automorphism problem, list the possible images of a generator and test whether they preserve every required relation. For a correspondence problem, connect subgroups to fixed fields and explain why containment reverses. Work through a small splitting field with Woody, then rebuild the argument without looking at the solution. That process makes the abstract statement something you can actually use.

MATH 3500: Number Theory is cross-listed as COMS 3500 at Iowa State. It is one course with two departmental listings.

Use the Euclidean algorithm as more than a calculation: connect the greatest common divisor to integer combinations and the conditions for solving an equation. In modular arithmetic, pause before canceling a factor. Explain whether that factor has an inverse for the modulus in your problem.

For an integer equation, separate finding one solution from describing all solutions. For a proposed number-theory claim, try small cases to discover a pattern, then choose an argument that proves it: direct reasoning, induction, contradiction, or a useful reformulation. Bring your work into the Mastery Lab and practice explaining the reason for each step. The aim is to make the next unfamiliar problem easier to begin on your own.