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is an antisymmetric matrix. The decomposition is also unique: a matrix that is both symmetric and antisymmetric must equal its own negative and hence vanish.

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 17 / b / ii by

Codex  0 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 17 / b / i by

Codex  0 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 17 / a / ii by

Codex  0 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 17 / a / i by

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 16 / b / Solution by

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Put

W = \nabla \times F

and

v = \dot{r}

. All vector fields and their spatial partial derivatives below are evaluated at

r (t)

. The product rule for a dot product gives

\frac{d}{d t} (W (r (t)) \cdot \dot{r} (t)) = \frac{d W ( r ( t ))}{d t} \cdot \dot{r} (t) + W (r (t)) \cdot \ddot{r} (t) .

(1)

Because the vector field has no explicit time dependence, the multivariable chain rule gives

d W / d t = J_{W} \dot{r}

, where the Jacobian matrix has entries

(J_{W})_{ij} = \partial_{j} W_{i}

. Consequently

\frac{d}{d t} [(\nabla \times F) \cdot \dot{r}] = \dot{r}^{T} J_{\nabla \times F} \dot{r} + (\nabla \times F) \cdot \ddot{r} .

(2)

The first term measures the change of the curl sampled as the point moves; the second measures the change in the velocity vector used in the dot product.

To give the result explicitly in

P, Q, R

, define the directional derivative operator

L = \overset{x}{˙} \partial_{x} + \overset{y}{˙} \partial_{y} + \overset{z}{˙} \partial_{z}

. Since

W = (R_{y} - Q_{z}, P_{z} - R_{x}, Q_{x} - P_{y})^{T},

(3)

the same expression is

\overset{x}{˙} L (R_{y} - Q_{z}) + \overset{y}{˙} L (P_{z} - R_{x}) + \overset{z}{˙} L (Q_{x} - P_{y}) + \overset{x}{¨} (R_{y} - Q_{z}) + \overset{y}{¨} (P_{z} - R_{x}) + \overset{z}{¨} (Q_{x} - P_{y}) .

(4)

For example

L (R_{y} - Q_{z}) = \overset{x}{˙} (R_{y x} - Q_{z x}) + \overset{y}{˙} (R_{yy} - Q_{zy}) + \overset{z}{˙} (R_{yz} - Q_{zz})

; the other two terms expand in the same way. Thus only the first and second spatial partial derivatives of

P, Q, R

, together with the velocity and acceleration of the parametric curve, occur.

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 16 / a / ii by

Codex  0 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 16 / a / i by

Codex  0 2026-10-18

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 15 / b / ii by

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 15 / b / i by

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 15 / a / Solution by

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Use the given Cartesian coordinates and integrate with respect to

y

first. Since

cos y e^{s i n y}

is the derivative of

e^{s i n y}

, the double integral becomes

I = \int_{0}^{π} cos x [e^{s i n y}]_{y = 0}^{y = x} d x = \int_{0}^{π} cos x (e^{s i n x} - 1) d x .

(1)

A direct antiderivative avoids any issue with the nonmonotonicity of

sin x

on this interval:

I = [e^{s i n x} - sin x]_{0}^{π} = 0 .

(2)

The sign changes in

cos x

make this cancellation possible even though

e^{s i n y}

is positive.

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 14 / b / Solution by

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Substitute the partial derivatives obtained above. The first-order terms cancel exactly:

u_{xx} - \frac{1}{2} u_{y} - y u_{yy} = U_{ξξ} + 2 U_{ξ η} + U_{ηη} - \frac{U _{ξ} - U _{η}}{2 y} - (U_{ξξ} - 2 U_{ξ η} + U_{ηη} - \frac{U _{ξ} - U _{η}}{2 y}) = 4 U_{ξ η} .

(1)

The transformed partial differential equation is therefore

U_{ξ η} = 0, ξ > η .

(2)

In these characteristic coordinates, the mixed partial derivative is the only surviving term.

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade

All our software is open source and hosted at: github.com/ourbigbook/ourbigbook

Further documentation can be found at: docs.ourbigbook.com

Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact

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