OpenLLM-France/wikimedia huggingface dataset:
# Moon/Quiz

Moon is a lecture about a rocky astronomical object in orbit around Earth.
This is a quiz based on lecture.
Your are free to take this quiz at any time.
To improve your score, study the lecture and its links: See also, External links and {{radiation astronomy resources}}.
Suggestion: Have the lecture available in a separate window.
Take the quiz as often as needed to improve your score, increase your familiarity with the subject, and improve your test-taking skills.
Enjoy learning by doing!
# Moodle/Sandbox server

Feel free to make use of the Wikiversity Moodle server for hands-on learning. No longer valid- it has been replaced by a spam/ad site. I'm not sure what the protocol is for removing this whole article, but it should be removed.

## Privacy issues

User:Draicone set up the Sandbox Server that is running the Wikiversity Moodle site. On 24 January 2008, Draicone made User:JWSchmidt and on 25 March 2008 User:Erkan Yilmaz administrators of the Wikiversity Moodle site. Moodle administrators can accessDraicone has not used the Moodle since then. JWSchmidt has CheckUser rights here at the English language Wikiversity. While the Moodle site is not officially part of Wikiversity, one option Wikimedia-like privacy would be to make User:SB Johnny (the other Wikiversity CheckUser) a Moodle administrator and demote Draicone from being an administrator.
    I'm happy for this adjustment to access permissions to be made, however it will not have any real effect as I would still have database access, as I administer the server in general. That said, I barely have time to check up on the Moodle site; I have never viewed a user's IP address either through moodle or directly through the database, and I don't even know where IP addresses are stored (although it wouldn't take long to find out). --Draicone $^{(talk)}$ 23:13, 25 March 2008 (UTC)

## Changes to default roles

In order to make the Wikiversity Moodle site more open to participation, an attempt has been made to give guests some -and authenticated users all- the abilities normally given by Moodle to students (Moodle roles). Suggest/discuss additional changes here:
# Literature/0000

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Wikimedia

| w: | w: |

## Chronology

* Literature// $^{[^]}$
* Literature// $^{[^]}$
* Literature// $^{[^]}$
* Literature// $^{[^]}$
* Literature// $^{[^]}$
# Literature/1800s/Mill

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &
    http://books.google.com/books?id=y4MEAAAAQAAJ

## Features

* In this work, he formulated the five principles of inductive reasoning that are known as Mill's methods.
* Extension and Intension
* Sense and reference
* Causal theory of reference

## Quotations

* A name is a word taken at pleasure to serve for a mark which may raise in our mind a thought like to some thought we had before, and which being pronounced to others may be to them a sign of what thought the speaker had before in his mind. (p. 27)‎
# Literature/1898/Wells

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &
* http://www.gutenberg.org/etext/36
* https://catalog.librivox.org/search.php?simple=The+War+of+the+Worlds (audiobook)

## Wikimedia

| w: Robert H. Goddard & w: Rocket w: Leo Szilard & w: Atomic bomb w: Vannevar Bush & w: Memex | w: T. H. Huxley w: Natural selection w: Social Darwinism |

### w: The War of the Worlds

* It ... influenced the real-life work of scientists, notably the rocket scientist Robert H. Goddard.
* Influential scientist Freeman Dyson, a key figure in the search for extraterrestrial life, also acknowledges his debt to reading H.G. Wells' fictions as a child.

## Chronology

* Wells, H. G. (1938). World Brain. Garden City, NY: Doubleday, Doran & Co. $^{[^]}$
    cf. w: Vannevar Bush
* Wells, H. G. (1933). The Shape of Things to Come. Hutchinson. $^{[^]}$
* Wells, H. G. (1928). The Open Conspiracy: Blue Prints For A World Revolution. Doubleday, Doran. $^{[^]}$
* Wells, H. G. (1923). Men Like Gods. Cassell and Co., Ltd. $^{[^]}$
* Wells, H. G. (1914). The World Set Free. Macmillan & Co. $^{[^]}$
    cf. w: Leo Szilard
* Wells, H. G. (1905). A Modern Utopia. New York, NY: Penguin Group, 2005. $^{[^]}$
* Wells, H. G. (1898). The War of the Worlds. William Heinemann. $^{[^]}$
    cf. w: Robert H. Goddard

## Reviews

* The War of the Worlds Invasion - containing comment and review on the history of The War of the Worlds.
# Literature/1900

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Subpages

* Literature/1900/Baum $^{[^]}$
* Literature/1900/Conrad $^{[^]}$
* Literature/1900/Freud $^{[^]}$
# Literature/1976/Halliday

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Reviews

http://www.goodreads.com/review/show/19633265
* This book changed linguistics. Before 1976 hardly anyone thought it was necessary to look beyond the sentence in order to study grammar. Without 'Cohesion in English' this erroneous view of language would probably still hold sway (and some like to think that they can still get away with it). Fortunately Halliday and Hasan were able to show how the grammatical system of reference works within and between sentences. This, of course, is just the beginning, although again many would be surprised to hear that Halliday and Hasan have moved on to bigger and better descriptions of English. Cohesion allows a description of language that extends up as far as the structure of each society but is all based on the same fundamental notion of meaning being created through choice.

## Wikimedia

| w: Context w: Psycholinguistics w: Sociolinguistics | w: Basil Bernstein w: Jan Mukarovsky w: Foregrounding |

## Related works

* Halliday, M.A.K. & Ruqaiya Hasan (1985). Language, Context and Text: Aspects of Language in a Social Semiotic Perspective. Deakin University Press & Oxford University Press. $^{[^]}$
* Halliday, M.A.K. & Ruqaiya Hasan (1976). Cohesion in English. London: Longman. $^{[^]}$
* Halliday, M.A.K. (1975). Learning How to Mean, London: Edward Arnold. $^{[^]}$
# Literature/1976/Miller

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Wikimedia

| w: Cognitive science w: Psycholinguistics | w: Mental model |

## Related works

* Philip N. Johnson-Laird (1983). Mental Models: Toward a Cognitive Science of Language, Inference and Consciousness. Harvard University Press. $^{[^]}$
* Gentner, Dedre & Albert L. Stevens, eds. (1983). Mental Models. Lawrence Erlbaum Associates. ISBN 0-89859-242-9. $^{[^]}$
* Johnson-Laird, Philip N. & Peter Cathcart Wason eds. (1977). Thinking: Readings in Cognitive Science. Cambridge University Press. $^{[^]}$
* Miller, George A. & Philip N. Johnson-Laird (1976). Language and Perception. Cambridge, MA: Belknap Press. $^{[^]}$
# Literature/1976/Pask

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Author
### w: Gordon Pask

   Pask held that concurrence is a necessary condition for modeling brain functions and he remarked IA was meant to stand AI, Artificial Intelligence, on its head. Pask believed it was the job of cybernetics to compare and contrast. His IA theory showed how to do this. Heinz von Foerster called him a genius, "Mr. Cybernetics", the "cybernetician's cybernetician".

## Wikimedia

| w: Second-order cybernetics | w: New Cybernetics (Gordon Pask) |

## Chronology

* Bateson, Gregory & Mead, Margaret (1976). "For God's Sake, Margaret: Conversation with Gregory Bateson and Margaret Mead." CoEvolutionary Quarterly (Summer 1976) no. 10, pp. 22-44. $^{[^]}$
* Pask, Gordon (1976). Conversation Theory: Applications in Education and Epistemology. New York: Elsevier. $^{[^]}$
* Pask, Gordon (1975). Conversation, Cognition and Learning. Elsevier. $^{[^]}$
* Pask, Gordon (1975). The Cybernetics of Human Learning and Performance. Hutchinson.

## Comments
### The inspiration of Conversation Theory

   Probably the most famous concept he introduced was 'Conversation Theory' which was the topic of the other book which he published in 1975 Conversation, Cognition and Learning and also of one he published in the following year Conversation Theory: Applications in Education and Epistemology. The initial ideas for Conversation Theory came from a slightly unlikely source, namely work that System Research undertook to assist baggage handling at London's Heathrow airport. He:-
       .... conceived human-machine interaction as a form of conversation, a dynamical process, in which the participants learn about each other.
   — http://www-history.mcs.st-and.ac.uk/Biographies/Pask.html
# Literature/1976/Warfield

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Wikimedia

| w: Cybernetics | w: System science |

## Chronology

* Warfield, John N. (1976). Societal Systems: Planning, Policy, and Complexity. New York: Wiley Interscience. $^{[^]}$
* Northrop, F. S. C. (1947). The Logic of the Sciences and the Humanities. New York: Meridian Books. $^{[^]}$ See also
* Bernal, J. D. (1939). The Social Function of Science. London: Routledge & Kegan Paul Ltd. $^{[^]}$
# Literature/1976/Weizenbaum

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Wikimedia
### w: Computer Power and Human Reason

* ... displays his ambivalence towards computer technology and lays out his case: while artificial intelligence may be possible, we should never allow computers to make important decisions because computers will always lack human qualities such as compassion and wisdom. Weizenbaum makes the crucial distinction between deciding and choosing. Deciding is a computational activity, something that can ultimately be programmed. It is the capacity to choose that ultimately makes us human. Choice, however, is the product of judgment, not calculation. Comprehensive human judgment is able to include non-mathematical factors such as emotions. Judgment can compare apples and oranges, and can do so without quantifying each fruit type and then reductively quantifying each to factors necessary for mathematical comparison.

### w: History of artificial intelligence #Critiques from across campus

* ... argued that the misuse of artificial intelligence has the potential to devalue human life.

### w: Ethics of artificial intelligence #The threat to human dignity
## Wikimedia

* w: Critique of technology

## Chronology

* Literature/1986/Dreyfus $^{[^]}$
* Literature/1983/Dreyfus $^{[^]}$
* Literature/1980/Searle $^{[^]}$
* Weizenbaum, Joseph (1976). Computer Power and Human Reason: From Judgment To Calculation. San Francisco: W. H. Freeman. $^{[^]}$
* Literature/1972/Dreyfus $^{[^]}$
* McCarthy, John; Marvin Minsky; Nathan Rochester & Claude Shannon (1955). A Proposal for the Dartmouth Summer Research Project on Artificial Intelligence. $^{[^]}$
# Literature/1977

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Subpages

* Literature/1977/Anderson $^{[^]}$
* Literature/1977/Arbib $^{[^]}$
* Literature/1977/Bandura $^{[^]}$
* Barnes, S. Barry (1977). Interests and the Growth of Knowledge. Routledge and Kegan Paul. $^{[^]}$
* Literature/1977/Blackmore $^{[^]}$
* Literature/1977/Bobrow $^{[^]}$
* Cohen, Jonathan (1977). The Probable and the Provable. Oxford University Press. $^{[^]}$
* Literature/1977/Dempster $^{[^]}$
* Literature/1977/Dick $^{[^]}$
* Literature/1977/Dijk $^{[^]}$
* Literature/1977/Dworkin $^{[^]}$
* Literature/1977/Eco $^{[^]}$
* Literature/1977/Foucault $^{[^]}$
* Gibson, Jame J. (1977). "The Theory of Affordances," pp. 67-82. In: Robert Shaw & John Bransford, eds. Perceiving, Acting, and Knowing: Toward an Ecological Psychology. Hillsdale, NJ: Lawrence Erlbaum. $^{[^]}$
* Literature/1977/Good $^{[^]}$
* Literature/1977/Hardwick $^{[^]}$
* Literature/1977/Harsanyi $^{[^]}$
* Literature/1977/Honderich $^{[^]}$
* Literature/1977/Hoskyns $^{[^]}$
* Literature/1977/Hutchins $^{[^]}$
* Johnson-Laird, Philip N. & Peter Cathcart Wason eds. (1977). Thinking: Readings in Cognitive Science. Cambridge University Press. $^{[^]}$
* Judge, Anthony (1977). "Knowledge-representation in a Computer-supported Ennvironment," International Classification, 4(2): 76-81. $^{[^]}$
* Literature/1977/Kaplan $^{[^]}$
* Literature/1977/Laudan $^{[^]}$
* Literature/1977/Lesk $^{[^]}$
* Literature/1977/Lorenz $^{[^]}$
* Literature/1977/Lyons $^{[^]}$
* Literature/1977/Maron $^{[^]}$
* Literature/1977/Moler $^{[^]}$
* Literature/1977/Morin $^{[^]}$
* Literature/1977/Narmour $^{[^]}$
* Literature/1977/Novak $^{[^]}$
* Literature/1977/Oddy $^{[^]}$
* Literature/1977/Piaget $^{[^]}$
* Popper, Karl and Eccles, John (1977). The Self and Its Brain: An Argument for Interactionism. Routledge, 1983. $^{[^]}$
* Literature/1977/Priest $^{[^]}$
* Literature/1977/Prigogine $^{[^]}$
* Literature/1977/Rescher $^{[^]}$
* Literature/1977/Rogers $^{[^]}$
* Literature/1977/Rosch $^{[^]}$
* Literature/1977/Schank $^{[^]}$
* Literature/1977/Schumacher $^{[^]}$
* Simon, Herbert (1977). Models of Discovery: And Other Topics in the Methods of Science. Springer $^{[^]}$
* Soergel, Dagobert (1977). "An Automated Encyclopedia: A Solution of the Information Problem?" International Classification, 4(1): 4-10; 4(2): 81-89. $^{[^]}$
* Literature/1977/Sternberg $^{[^]}$
* Literature/1977/Temkin $^{[^]}$
* Literature/1977/Tukey $^{[^]}$
* Literature/1977/Turchin $^{[^]}$
* Literature/1977/Turner $^{[^]}$
* Literature/1977/Waddington $^{[^]}$
* Wilson, Patrick (1977). Public Knowledge, Private Ignorance: Toward a Library and Information Policy. Greenwood Publishing Group. $^{[^]}$
# Literature/1976/Hampshire

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &
# Literature/1976/Neisser

Authors :
* A B C D E F G H I J K L M N O P Q R S T U V W X Y Z &

## Wikimedia
### w: Ulric Neisser

* In 1976, he wrote Cognition and Reality, in which he expressed three general criticisms of the field of cognitive psychology. First, he was dissatisfied with the linear programming model of cognitive psychology, with its over-emphasis on peculiar information processing models used to describe and explain behavior. Second, he felt that cognitive psychology had failed to address the everyday aspects and functions of human behavior. He placed blame for this failure largely on the excessive reliance on artificial laboratory tasks that had become endemic to cognitive psychology by the mid-1970s. In this sense, he felt that cognitive psychology suffered a severe disconnect between theories of behavior tested by laboratory experimentation and real-world behavior, which he called a lack of ecological validity. Lastly, and perhaps most importantly, he had come to feel a great respect for the theory of direct perception and information pickup that had been promulgated by the preeminent perceptual psychologist J. J. Gibson and his wife, the "grand dame" of developmental psychology, Eleanor Gibson. Neisser, in this book, had come to the conclusion that cognitive psychology had little hope of achieving its potential without taking careful theoretical note of the Gibsons' work on perception which argued that understanding human behavior first involves careful analysis of the information available to any perceiving organism.
| w: Cognitive psychology | w: Cognitive science |

## Chronology

* Literature/1980/Searle $^{[^]}$
* Gibson, Jame J. (1977). "The Theory of Affordances," pp. 67-82. In: Robert Shaw & John Bransford, eds. Perceiving, Acting, and Knowing: Toward an Ecological Psychology. Hillsdale, NJ: Lawrence Erlbaum. $^{[^]}$
* Chisholm, Roderick (1976). Person and Object: A Metaphysical Study. London: G. Allen & Unwin. $^{[^]}$
* Leach, Edmund (1976). Culture and Communication: The Logic by Which Symbols are Connected. Cambridge University Press. $^{[^]}$
* Neisser, Ulric (1976). Cognition and Reality: Principles and Implications of Cognitive Psychology. WH Freeman. $^{[^]}$
* Weizenbaum, Joseph (1976). Computer Power and Human Reason: From Judgment To Calculation. San Francisco: W. H. Freeman. $^{[^]}$
* Fodor, Jerry (1975). The Language of Thought. Harvard University Press. $^{[^]}$
* Neisser, Ulric (1967). Cognitive Psychology. Prentice Hall. $^{[^]}$
* Skinner, B. F. (1957). Verbal Behavior. Acton, Massachusetts: Copley Publishing Group. $^{[^]}$
* Craik, Kenneth (1943). The Nature of Explanation. Cambridge University Press. $^{[^]}$
* Moore, Thomas Verner (1939). Cognitive Psychology. J. B. Lippincott Co. $^{[^]}$
* Moore, Thomas Verner (1924). Dynamic Psychology: An Introduction to Modern Psychological Theory and Practice. J. B. Lippincott Co. $^{[^]}$
* Ogden, C. K. & I. A. Richards (1923). The Meaning of Meaning: A Study of the Influence of Language upon Thought and of the Science of Symbolism. London: Routledge & Kegan Paul Ltd. $^{[^]}$
# Micromechanics of composites/Proof 1
## Tensor-vector identity - 1

    $[(\mathbf {v} \bullet \mathbf {a} )({\boldsymbol {S}}\bullet \mathbf {b} )]\cdot \mathbf {n} =\mathbf {a} \cdot [\{\mathbf {v} \otimes ({\boldsymbol {S}}^{T}\bullet \mathbf {n} )\}\cdot \mathbf {b} ]~.$
Proof:
Using the identity $\mathbf {a} \cdot ({\boldsymbol {A}}^{T}\cdot \mathbf {b} )=\mathbf {b} \cdot ({\boldsymbol {A}}\cdot \mathbf {a} )$ we have
    $\mathbf {n} \cdot [(\mathbf {v} \bullet \mathbf {a} )({\boldsymbol {S}}\bullet \mathbf {b} )]=\mathbf {b} \cdot [(\mathbf {v} \cdot \mathbf {a} )({\boldsymbol {S}}^{T}\cdot \mathbf {n} )]~.$
Also, using the definition $(\mathbf {u} \otimes \mathbf {v} )\cdot \mathbf {a} =(\mathbf {a} \cdot \mathbf {v} )\mathbf {u}$ we have
    $(\mathbf {v} \cdot \mathbf {a} )({\boldsymbol {S}}^{T}\cdot \mathbf {n} )=[({\boldsymbol {S}}^{T}\cdot \mathbf {n} )\otimes \mathbf {v} ]\cdot \mathbf {a} ~.$
Therefore,
    $\mathbf {n} \cdot [(\mathbf {v} \bullet \mathbf {a} )({\boldsymbol {S}}\bullet \mathbf {b} )]=\mathbf {b} \cdot [\{({\boldsymbol {S}}^{T}\cdot \mathbf {n} )\otimes \mathbf {v} \}\cdot \mathbf {a} ]~.$
Using the identity $\mathbf {a} \cdot ({\boldsymbol {A}}^{T}\cdot \mathbf {b} )=\mathbf {b} \cdot ({\boldsymbol {A}}\cdot \mathbf {a} )$ we have
    $\mathbf {b} \cdot [\{({\boldsymbol {S}}^{T}\cdot \mathbf {n} )\otimes \mathbf {v} \}\cdot \mathbf {a} ]=\mathbf {a} \cdot [\{({\boldsymbol {S}}^{T}\cdot \mathbf {n} )\otimes \mathbf {v} \}^{T}\cdot \mathbf {b} ]~.$
Finally, using the relation $(\mathbf {u} \otimes \mathbf {v} )^{T}=\mathbf {v} \otimes \mathbf {u}$, we get
    $\mathbf {a} \cdot [\{({\boldsymbol {S}}^{T}\cdot \mathbf {n} )\otimes \mathbf {v} \}^{T}\cdot \mathbf {b} ]=\mathbf {a} \cdot [\{\mathbf {v} \otimes ({\boldsymbol {S}}^{T}\cdot \mathbf {n} )\}\cdot \mathbf {b} ]~.$
Hence,
    ${[(\mathbf {v} \bullet \mathbf {a} )({\boldsymbol {S}}\bullet \mathbf {b} )]\cdot \mathbf {n} =\mathbf {a} \cdot [\{\mathbf {v} \otimes ({\boldsymbol {S}}^{T}\bullet \mathbf {n} )\}\cdot \mathbf {b} ]}\qquad \qquad \qquad \square$
# Micromechanics of composites/Average velocity gradient in a RVE
## Average velocity gradient in a RVE

The time rate of the deformation gradient is given by
    ${\dot {\boldsymbol {F}}}={\frac {\partial }{\partial t}}[{\boldsymbol {F}}(\mathbf {X} ,t)]={\frac {\partial }{\partial t}}\left({\frac {\partial }{\partial \mathbf {X} }}[\mathbf {x} (\mathbf {X} ,t)]\right)={\frac {\partial }{\partial \mathbf {X} }}\left({\frac {\partial }{\partial t}}[\mathbf {x} (\mathbf {X} ,t)]\right)={\frac {\partial {\dot {\mathbf {x} }}}{\partial \mathbf {X} }}={\boldsymbol {\nabla }}_{0}~{\dot {\mathbf {x} }}~.$
The average time rate of the deformation gradient is defined as
    ${\langle {\dot {\boldsymbol {F}}}\rangle :={\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}~{\text{dV}}~.}$
Following the same procedure as in the previous section, we can show that
    ${\langle {\dot {\boldsymbol {F}}}\rangle ={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes \mathbf {N} ~{\text{dA}}={\cfrac {1}{V}}\int _{\partial {\Omega }}({\dot {\mathbf {x} }}\otimes \mathbf {n} )\cdot {\boldsymbol {F}}~{\text{da}}~.}$
The velocity gradient (${\boldsymbol {l}}$) is given by
    ${\boldsymbol {l}}={\boldsymbol {\nabla }}\mathbf {v} ={\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}$
where $\mathbf {v} (\mathbf {x} )$ is the velocity.
The average velocity gradient in a RVE is defined as
    ${{\overline {\boldsymbol {l}}}:=\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {F}}\rangle ^{-1}~.}$
Note that ${\overline {\boldsymbol {l}}}=\langle {\boldsymbol {l}}\rangle =\langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}\rangle$ only if ${\boldsymbol {F}}={\boldsymbol {\mathit {1}}}$.
# Micromechanics of composites/Conservation of mass
## Statement of the balance of mass

The balance of mass can be expressed as:
where $\rho (\mathbf {x} ,t)$ is the current mass density, ${\dot {\rho }}$ is the material time derivative of $\rho$, and $\mathbf {v} (\mathbf {x} ,t)$ is the velocity of physical particles in the body $\Omega$ bounded by the surface $\partial {\Omega }$.

## Proof

We can show how this relation is derived by recalling that the general equation for the balance of a physical quantity $f(\mathbf {x} ,t)$ is given by
    ${\cfrac {d}{dt}}\left[\int _{\Omega }f(\mathbf {x} ,t)~{\text{dV}}\right]=\int _{\partial {\Omega }}f(\mathbf {x} ,t)[u_{n}(\mathbf {x} ,t)-\mathbf {v} (\mathbf {x} ,t)\cdot \mathbf {n} (\mathbf {x} ,t)]~{\text{dA}}+\int _{\partial {\Omega }}g(\mathbf {x} ,t)~{\text{dA}}+\int _{\Omega }h(\mathbf {x} ,t)~{\text{dV}}~.$
To derive the equation for the balance of mass, we assume that the physical quantity of interest is the mass density $\rho (\mathbf {x} ,t)$. Since mass is neither created or destroyed, the surface and interior sources are zero, i.e., $g(\mathbf {x} ,t)=h(\mathbf {x} ,t)=0$. Therefore, we have
    ${\cfrac {d}{dt}}\left[\int _{\Omega }\rho (\mathbf {x} ,t)~{\text{dV}}\right]=\int _{\partial {\Omega }}\rho (\mathbf {x} ,t)[u_{n}(\mathbf {x} ,t)-\mathbf {v} (\mathbf {x} ,t)\cdot \mathbf {n} (\mathbf {x} ,t)]~{\text{dA}}~.$
Let us assume that the volume $\Omega$ is a control volume (i.e., it does not change with time). Then the surface $\partial {\Omega }$ has a zero velocity ($u_{n}=0$) and we get
    $\int _{\Omega }{\frac {\partial \rho }{\partial t}}~{\text{dV}}=-\int _{\partial {\Omega }}\rho ~(\mathbf {v} \cdot \mathbf {n} )~{\text{dA}}~.$
Using the divergence theorem
    $\int _{\Omega }{\boldsymbol {\nabla }}\bullet \mathbf {v} ~{\text{dV}}=\int _{\partial {\Omega }}\mathbf {v} \cdot \mathbf {n} ~{\text{dA}}$
we get
    $\int _{\Omega }{\frac {\partial \rho }{\partial t}}~{\text{dV}}=-\int _{\Omega }{\boldsymbol {\nabla }}\bullet (\rho ~\mathbf {v} )~{\text{dV}}.$
or,
    $\int _{\Omega }\left[{\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\bullet (\rho ~\mathbf {v} )\right]~{\text{dV}}=0~.$
Since $\Omega$ is arbitrary, we must have
    ${\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\bullet (\rho ~\mathbf {v} )=0~.$
Using the identity
    ${\boldsymbol {\nabla }}\bullet (\varphi ~\mathbf {v} )=\varphi ~{\boldsymbol {\nabla }}\bullet \mathbf {v} +{\boldsymbol {\nabla }}\varphi \cdot \mathbf {v}$
we have
    ${\frac {\partial \rho }{\partial t}}+\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} +{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} =0~.$
Now, the material time derivative of $\rho$ is defined as
    ${\dot {\rho }}={\frac {\partial \rho }{\partial t}}+{\boldsymbol {\nabla }}\rho \cdot \mathbf {v} ~.$
Therefore,
    ${{\dot {\rho }}+\rho ~{\boldsymbol {\nabla }}\bullet \mathbf {v} =0~.}$