Math · 06
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
For \(f(x,y)=x^2+y^2\), compute the gradient at \((1,1)\) and say what it points toward.
Timer not started. Twenty minutes of real work before a solution.
\(\nabla f=(2x,2y)\), so at \((1,1)\) it is \((2,2)\). It points in the direction of steepest increase, and its length is that slope.
Attack 2
Compute both mixed second partials of \(f(x,y)=x^2 y+y^3\).
Timer not started. Twenty minutes of real work before a solution.
\(f_x=2xy\), \(f_y=x^2+3y^2\), \(f_{xy}=2x\), \(f_{yx}=2x\). They agree, as Clairaut’s theorem predicts when the second partials are continuous.
Attack 3
Why is the line integral of a gradient around a closed loop equal to 0?
Timer not started. Twenty minutes of real work before a solution.
If \(\mathbf F=\nabla f\), then along a path the integral is the telescope \(f(\text{end})-f(\text{start})\) by the chain rule and the single-variable fundamental theorem. On a closed loop those points agree, so the integral is 0. The domain has to make the fundamental theorem applicable along the path.
Multivariable calculus is single-variable calculus with linear algebra inserted at every derivative. The derivative of a map \(\mathbb{R}^n\to\mathbb{R}^m\) at a point is the linear map that best approximates it there, represented by the Jacobian matrix. Gradients, directional derivatives, chain rules, and tangent planes are that one definition specialized. Integration becomes a limit of sums over boxes, and the change-of-variables theorem multiplies by the absolute value of the Jacobian determinant, which is why determinants were worth learning.
The second half is the integral theorems: Green, Stokes, and the divergence theorem. They are one theorem, the generalized Stokes theorem, cut into the cases a first course can compute. Schey’s Div, Grad, Curl, and All That (ISBN 978-0-393-92516-6, paid) is the short intuition book; the author is H. M. Schey, not Saff and Snider. Marsden and Tromba, Vector Calculus, ISBN 978-1-4292-1508-4, is a standard paid text. You will work from MIT 18.02 and OpenStax Volume 3, which are free and have exercises.
Learn Anything has no separate vetted path here beyond the calculus junk drawer. The multivariable items in that drawer were course fragments. 18.02 is the course.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Derivative as a linear map. \(f:\mathbb{R}^n\to\mathbb{R}^m\) is differentiable at \(a\) when there is a linear map \(Df(a)\) with \(f(a+h)=f(a)+Df(a)(h)+o(\|h\|)\). For \(m=1\) the matrix of \(Df\) is the gradient row. The chain rule is composition of those linear maps.
In coordinates this is the existence of partial derivatives plus a little theorem: continuous partials imply differentiability. The chain rule is the chain rule for the linear approximations, with an error that is still \(o(\|h\|)\).
Extrema. At an interior local extremum of a differentiable function, the gradient vanishes. On a constraint \(g=c\) with \(\nabla g\ne 0\), Lagrange multipliers say \(\nabla f=\lambda\nabla g\). The Hessian test classifies nondegenerate critical points.
Restrict to lines through the point and apply the single-variable Fermat theorem; every directional derivative vanishes, hence the gradient does. Lagrange is the statement that \(\nabla f\) is normal to the level set, and \(\nabla g\) spans that normal line.
Multiple integrals. A continuous function on a rectangle is Riemann integrable. Fubini’s theorem reduces an integral over a rectangle to iterated one-variable integrals. Under a \(C^1\) change of variables with nonzero Jacobian, \(dx\,dy=|\det D\Phi|\,du\,dv\).
Fubini for continuous functions is an ε/3 argument with uniform continuity on a compact rectangle. The change-of-variables factor is the area distortion of the derivative linear map, which is the absolute value of the determinant, integrated because the derivative varies from point to point.
Vector calculus identities. \(\operatorname{curl}\nabla f=0\) and \(\operatorname{div}\operatorname{curl}\mathbf F=0\) wherever the derivatives exist and are continuous. A \(C^1\) vector field on a simply connected domain in \(\mathbb{R}^3\) is conservative if and only if its curl is zero.
The two identities are equality of mixed partials. The conservative-field theorem in the plane is Green’s theorem plus the curl test; in space it is Stokes plus a disk spanned by the loop, which is where simple connectedness is used.
Green, Stokes, divergence. Green: \(\int_{\partial D} P\,dx+Q\,dy=\iint_D(Q_x-P_y)\,dA\). Stokes: \(\int_{\partial S}\mathbf F\cdot d\mathbf r=\iint_S(\nabla\times\mathbf F)\cdot d\mathbf S\). Divergence: \(\iint_{\partial W}\mathbf F\cdot d\mathbf S=\iiint_W\nabla\cdot\mathbf F\,dV\). Orientations match.
Green is proved for a rectangle by writing each side as an iterated integral, then extended by cutting a nice region into pieces. Stokes and the divergence theorem reduce to Green in coordinates, or they are quoted as the cases of the general Stokes theorem you will see properly in a manifolds course. For this page, be able to prove Green on a rectangle.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
Paul’s Online Math Notes, Calculus III
Read once, do not study from this. Use the section list as a map of topics. The notes are a reference you already know how to find, not the course.
The thing you work through.
MIT 18.02SC and OpenStax Calculus Volume 3
18.02 is the lecture and exam path. OpenStax Volume 3 is the free book: https://openstax.org/details/books/calculus-volume-3 . Marsden and Tromba, ISBN 978-1-4292-1508-4, is the paid text if you want a tighter one. Schey, ISBN 978-0-393-92516-6, is the short paid supplement for the field theorems.
The problem book.
18.02 problem sets and Apostol, Volume 2
Do the 18.02 sets before you shop. Apostol, Calculus, Volume 2, is the paid harder text; the Volume 1 ISBN was verified, the Volume 2 ISBN was not, so it is named without a number.
The hardest questions that exist at this level.
18.02 exams and Putnam problems Kedlaya files under geometry or analysis
The exam page is the same 18.02 site. The Putnam archive is the public ceiling. Borisov’s sorted list helps you find the vector-calculus problems: https://people.math.binghamton.edu/borisov/Putnam/SelectedProblemList.pdf
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: one 18.02 exam, timed, plus written proofs of Green’s theorem on a rectangle and of the fact that curl of a gradient is zero. Fail if you cannot parametrize a sphere or state the hypotheses of Stokes.
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
Define the derivative of a function of several variables, state Lagrange multipliers, and state Stokes’ theorem with orientations. Blank page.