π Multivariable Calculus
Math
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This deck contains everything taught in the Multivariable Calculus course on Khan Academy with additional material from Calculus (8th ed) by Stββewβββart such that this deck covers literally everything.
Download the version of this deck that has images for free on kofi.
βοΈ Features βοΈ
- Cards in the deck contain plentiful derivations, proofs, images, and context on the back so that you understand where formulas come from.
- Every card is color-coded and math is written in MathJax
- Every card includes a link to the respective videos and articles on Khan Academy and are thoroughly tagged by their unit & topic.
The cards in this deck work with the Clickable Tags addon. - All cards are ordered so that material that comes earlier in the course shows up as new cards before material that comes later.
- "Exercise" cards where you can practice applying the formulas you memorized, using spaced repetition to bolster your procedural memory.
π Contents of the deck, organized by tags π:
βοΈ Prerequisites for the course and deck π
- AP Calculus BC
- Vectors and matrices
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Sample Data
| Text | A curve or surface is {{c1::piecewise-smooth::property}} if it {{c2::can be broken up into a finite number of smooth differentiable parts::condition}}. |
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| Summary 1 | Types of curves |
| Back Extra | Here's an example of a simple, closed, piecewise-smooth curve:
 Here's an example of a curve that's not simple because it intersects itself:
 And here's an example of a curve that's not simple nor closed because it intersects itself and starts and ends at different points:
 Nearly all functions we deal with are piecewise smooth. If you want more details about functions that aren't piecewise-smooth, look into something called the Weierstrass function. |
| Summary 2 | Stoke's theorem and conditions |
| Back Extra 2 | Stokes' theorem states that for a function \(\vec{\mathbf F}(x,y,z)=(P,Q,R)\),\(\large\oint_C \vec {\mathbf F} \cdot d\vec {\mathbf r}\) \(\large=\quad\) \(\large \iint_S (\text{curl }\vec {\mathbf F})\cdot \hat{n} \, dA\)Where \(S\) is a surface that is piecewise-smooth, bounded by a curve \(C\) that is simple, closed, and piecewise-smooth. |
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| Text | A region is {{c1::simply connected::property}} if it {{c2::never intersects itself and it has no holes::condition}}. |
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| Summary 1 | For comparison |
| Back Extra | A curve is simple if it never intersects itself. |
| Summary 2 | Types of regions |
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 More rigorously, a region \(D\) is simply connected if we can take any curve contained in \(D\) and continuously shrink it until it's a single point, always requiring that the curve stays in \(D\). That means that infinite regions can also be simply connected. The rigorous definition of continuously shrinking comes from topology.Here's an example of a simply connected region:
 Here's an example of a region that's not simply connected because it intersects itself:
 And here's an example of a region that's not simply connected because it has a hole:
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| Summary 3 | Stoke's theorem and conditions |
| Back Extra 3 | Stokes' theorem states that for a 3d vector field \(\vec{\mathbf F}(x,y,z)=(P,Q,R)\),\(\large\oint_C \vec {\mathbf F} \cdot d\vec {\mathbf r}\) \(\large=\quad\) \(\large \iint_S (\text{curl }\vec {\mathbf F})\cdot \hat{n} \, dA\)Where \(S\) is a surface that is piecewise-smooth, bounded by a curve \(C\) that is simple, closed, and piecewise-smooth. |
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| Text | {{c1::Green's theorem::theorem}} states that for a 2d vector field \(\vec{\mathbf F}(x,y)=(P, Q)\),{{c2::\(\begin{align}\oint_C \vec {\mathbf F} \cdot d\vec {\mathbf s}\end{align}\)::value}} \(\large=\) {{c2::\(-\begin{align} \iint_R \text{2d-curl }\vec {\mathbf F} \, dA\end{align}\)::value}}Where \(R\) is some region in the \(xy\)-plane, bounded by a {{c1::negatively::positively or negatively}} oriented curve \(C\), meaning that it is circulated {{c1::clockwise::direction}}. |
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| Summary 1 | Info |
| Back Extra | Recall that positive 2d-curl implies that the imaginary fluid directed by the vector field is flowing in a counterclockwise direction. So if we circulate \(C\) in the clockwise direction, then we'll have to multiply the result by \(-1\).If the curve is circulated counterclockwise, then the curve is positively oriented and the equation becomes \(\oint_C \vec {\mathbf F} \cdot d\vec {\mathbf s}\) \(=\quad\) \(\iint_R \text{2d-curl }\vec {\mathbf F} \, dA\)
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| Summary 2 | Derivation (intuition) of Green's theorem |
| Back Extra 2 | Imagine chopping up a region into 2 regions. The sum of the line integral around the boundary of each region is just the line integral around the boundary of the whole region (the middle part is in opposite directions and cancels out)\[\begin{equation}\oint_{C_1} \mathbf{F} \cdot d\mathbf{s} + \oint_{C_2} \mathbf{F} \cdot d\mathbf{s} = \oint_{C} \mathbf{F} \cdot d\mathbf{s}\end{equation}\]
 If we continue to split the regions up...
 The line integral around the boundary of each component part is \[\begin{equation}\oint_{C_k} \mathbf{F} \cdot d\mathbf{s} \approx \left( \underbrace{\text{2d-curl} \ \mathbf{F} (x_k, y_k)}_{\text{Point in } R_k} \right) \underbrace{|R_k|}_{\text{Area of } R_k}\end{equation}\]In fact, the formal definition of \(\text{2d-curl } \mathbf{F}(x, y) = \lim_{|R(x,y)| \to 0} \left( \frac{1}{|R(x,y)|} \oint_{C(x,y)} \mathbf{F} \cdot d\mathbf{s} \right)\).
 Adding an infinite of these infinitesimally small components of the line integral up, we get \(\large\oint_C \vec {\mathbf F} \cdot d\vec {\mathbf s}\) \(\large=\) \(\large \iint_R \text{2d-curl }\vec {\mathbf F} \, dA\) |
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