Concordia University · MAST 219

Concordia MAST 219 tutoring

Multivariable Calculus II: one-to-one online tutoring and exam preparation.

Online 60 min: $55 CAD · Pricing Midterm & final exam help

Listed course

Course overview

MAST 219 develops multiple integration and vector calculus, including integrals over curves and surfaces. The closed-surface example below focuses on outward flux; use your section outline to place it within the course. Send your chapter and attempt so we can confirm support before booking.

MAST 219 topics to review

Double and triple integralsChanges of coordinates and variablesLine integrals and conservative fieldsSurface integrals and orientationGreen, Stokes and divergence theorems

Common difficulties

Checking whether a surface is closed
Choosing consistent outward normals
Distinguishing net flux from individual face contributions

Check the closed surface before using divergence

All coordinates and field values in this mathematical model are dimensionless. Let $E=[0,1]\times[0,2]\times[0,1]$ be a solid box and $S=\partial E$ its entire boundary, including all six faces. For $\mathbf F(x,y,z)=(x,2y,3z)$ , find the outward flux $\Phi=\iint_S\mathbf F\cdot\mathbf n\,dS$ , where $\mathbf n$ is the outward unit normal and $dS$ is a surface-area element. Predict the sign before integrating.

The arrows are outward normal directions, not values of the field. Include the hidden faces too.

Hint: test one face first

On $x=1$ , the outward normal is $(1,0,0)$ , so only the first component of the field contributes. The divergence theorem applies to a closed, piecewise-smooth boundary when the field has continuous first partial derivatives on an open set containing the solid.

Compare faces with the volume calculation

The polynomial field is continuously differentiable everywhere and the box has a closed piecewise-smooth boundary. Its divergence is $\nabla\cdot\mathbf F=1+2+3=6$ . The volume is $1\cdot2\cdot1=2$ , so:

\Phi=\iiint_E(\nabla\cdot\mathbf F)\,dV=6(2)=12.

For a direct check, the $x=1$ face contributes $1\times2=2$ , the $y=2$ face contributes $4\times1=4$ , and the $z=1$ face contributes $3\times2=6$ . Each coordinate-zero face contributes zero because its corresponding field component vanishes. The sum is $2+4+6=12$ , consistent with positive outward flux.

Do not use the volume formula for just the top face: the theorem gives the flux through the whole closed boundary. A face normal is not automatically the direction of the field.

Retry: can zero total flux hide nonzero face flux?

Keep the box and outward orientation, but use $\mathbf G=(x,2y,-3z)$ . Find the total flux and the top-face contribution. Does zero total flux mean the field is zero on the boundary?

Check the changed field

Now $\nabla\cdot\mathbf G=1+2-3=0$ , so the total outward flux is zero. The top face has $\mathbf G\cdot\mathbf n=-3$ and area $2$ , giving $-6$ . The other nonzero contributions are $2$ and $4$ . They cancel: $2+4-6=0$ . Zero net flux does not mean zero field or zero flux on every face.

If orientation itself is unclear, use the existing path-orientation refresher. It concerns curves; the surface-normal check above is a separate step.

Move from a closed boundary to an open surface

The box example above includes all six faces. If your next problem gives an open curved surface, mark its edge and decide what would close it. Use the surface-flux lesson to build a signed area vector, then try the six guided flux questions. Compare a direct calculation with a closing-surface calculation, keeping each added contribution signed.

For multiple-integral bounds rather than a normal-direction problem, use the existing double-integral questions. Bring your sketch, included surfaces and first unsupported step to a tutor.

How tutoring can help

Bring your surface sketch and one face calculation. Your tutor can check which faces are included, the outward normals and the field’s regularity before using a theorem. Compare one direct contribution with the volume calculation, then change the field and predict which terms reverse sign.

Exam preparation for MAST 219

Practise setting bounds for multiple integrals and choosing between direct line or surface integration and the appropriate theorem. Check the region, orientation and field assumptions before calculating, then explain how the result matches the geometry.

Use your section’s outline to choose the chapters to review. We work through permitted practice and comparable problems, without completing graded assignments or exams on your behalf.

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MAST 219 · official course reference MAST 219 · official course outline