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Stokes' theorem proof part 1 | Multivariable Calculus | Khan Academy

Video vocabulary

try

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US / traɪ /

UK / traɪ /

A1
  • noun
  • Act of putting the ball over the rugby goal line
  • verb
  • To make an effort, to attempt to do something
  • To examine someone or something in a court of law
  • To test to see if it is good or you like it
  • To make a person go through hardship; test
  • To examine something to see if it works

after

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US / 'ɑ:ftə(r) /

UK / ˈæftɚ /

A1
  • adverb
  • Later than another time, or behind something
  • conjunction
  • Following something before
  • preposition
  • When an event happened or time has passed

write

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US / raɪt /

UK / raɪt /

A1
  • verb
  • To compose letters and words on paper or a screen
  • To invent or create a computer program
  • To create a song or piece of music

will

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US / wɪl /

UK / wɪl /

A1
  • noun
  • Desire or choice of someone
  • Document saying who gets your money when you die
  • Strong desire to do something; determination
  • Desire to do something; strong wish to do
  • other
  • Used with verbs to express the future
  • verb
  • To leave property to someone after your death
  • To use mental effort to make something happen
  • Am (is, are) likely to do
  • To influence someone to do something
  • To make something happen by strongly wanting it

here

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US / hɪə(r) /

UK / hɪr /

A1
  • adverb
  • In this place; at this location, position or time

integral

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US / ˈɪntɪgrəl /

UK / ˈɪntɪɡrəl, ɪnˈtɛɡrəl /

B1
  • adjecitve
  • Being an essential part of something; necessary

equivalent

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US / ɪˈkwɪvələnt /

UK / ɪˈkwɪvələnt /

B1
  • adjecitve
  • Equal to something in value, use or meaning
  • noun
  • Thing like another in quality, quantity or degree

probably

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US / ˈprɒbəbli /

UK / ˈprɑbəbli /

A1
  • adverb
  • That is likely to happen or be true

point

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US / pɔɪnt /

UK / pɔɪnt /

A1
  • noun
  • An item to be discussed
  • Small spot or dot
  • Fact or detail
  • Feature of something
  • The size of the text on a page or screen
  • A certain time or stage in a process
  • Certain position in time or space
  • Idea or fact you try to convince people about
  • Measurement of scores in a game, sport
  • Sharp end of something, e.g. a pen or pin
  • verb
  • To face a certain direction, e.g. north
  • To indicate something with your finger to others

should

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US / ʃʊd /

UK / ʃʊd /

A1
  • other
  • Used to indicate what is proper or reasonable

plus

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US / plʌs /

UK / plʌs /

A2
  • noun
  • A benefit or advantage
  • preposition
  • In addition; add
  • In addition to; Indicates a number above
  • Having a value greater than zero

actually

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US / ˈæktʃuəli /

UK / ˈæktʃuəli /

A1
  • adverb
  • Used to add new (often different) information

component

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US / kəmˈpəʊnənt /

UK / kəmˈponənt /

A2
  • adjecitve
  • Being a part of something
  • noun
  • One of the parts that something is made up of

reason

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US / 'ri:zn /

UK / ˈrizən /

A1
  • noun
  • Ability to think about facts and form a judgment
  • Explanation for why something occurred or was done
  • verb
  • To think and make conclusions in a logical manner

dimension

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US / daɪˈmenʃn /

UK / dɪˈmɛnʃən, daɪ- /

B1
  • noun
  • Number of planes (X, Y, Z axes) an object has
  • Measurement in space, e.g. length or diameter
  • Scope, extent or size of something abstract

itself

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US / ɪtˈself /

UK / ɪtˈsɛlf /

A1
  • pronoun
  • Referring to it

guess

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US / ɡes /

UK / ɡɛs /

A1
  • noun
  • Answer given when you don't know if it is correct
  • verb
  • To give an answer without knowing if it is correct

region

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US / 'ri:dʒən /

UK / ˈridʒən /

A2
  • noun
  • Part of a country, of the world, area, etc.

continuous

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US / kənˈtɪnjuəs /

UK / kənˈtɪnjuəs /

A2
  • adjecitve
  • Happening, existing, or performed without stops

height

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US / haɪt /

UK / haɪt /

A2
  • noun
  • Distance of something from the bottom to the top

something

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US / ˈsʌmθɪŋ /

UK / ˈsʌmθɪŋ /

A1
  • adverb
  • In some degree or measure
  • noun
  • a person or thing of some value or consequence
  • pronoun
  • Thing that is not yet known or named
  • Approximately

think

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US / θɪŋk /

UK / θɪŋk /

A1
  • verb
  • To have an idea about something without certainty
  • To have an idea, opinion or belief about something

cross

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US / krɒs /

UK / krɔs, krɑs /

A2
  • adjecitve
  • Angry and upset
  • From one side of something to the other
  • noun
  • (Of animals) mixture of breeds in one animal
  • Person's name
  • verb
  • To put one thing over another e.g. arms
  • To breed one type of animal or plant with another
  • To not to agree with someone; oppose
  • To go from one side of something to the other
  • To meet at one point

two

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US / tu: /

UK / tu /

A1
  • number
  • 2

particular

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US / pəˈtɪkjələ(r) /

UK / pɚˈtɪkjəlɚ, pəˈtɪk- /

B2
  • adjecitve
  • One specific (one)
  • Having firm ideas about what is desirable
  • Being special or more than is typical

out

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US / aʊt /

UK / aʊt /

A1
  • adverb
  • No longer stylish (e.g. fashion)
  • Away from home, business, or the usual place
  • In a direction/movement away from inside or center
  • noun
  • Act when a batter cannot continue playing
  • verb
  • To tell people a secret

only

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US / ˈəunli /

UK / ˈonli /

A1
  • adverb
  • Just one; just that amount or thing; not more than
  • Not very important or serious
  • Just that type of thing, person
  • Short time ago; not before a certain time
  • conjunction
  • Except; but

any

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US / 'enɪ /

UK / ˈɛni /

A1
  • adjecitve
  • One, or some, no matter which
  • pronoun
  • One (thing) of many; some

version

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US / ˈvə:ʃən /

UK / ˈvɚʒən, -ʃən /

A2
  • noun
  • Different way that someone interprets something
  • New or different form of something
  • An updated form of some software

finally

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US / ˈfaɪnəli /

UK / ˈfaɪnəli /

A1
  • adverb
  • used especially at the beginning of a sentence to introduce the last point or idea
  • After a long time or some difficulty

proof

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US / pru:f /

UK / pruf /

A2
  • noun
  • Act of showing that something exists or is true
  • Version of a document to be checked for errors
  • Special coating that protects something
  • Something which shows that something else is true
  • verb
  • To read a written work to check for mistakes
  • To protect by painting on a special coating

where

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US / hwɛə /

UK / hwɛr, wɛr /

A1
  • conjunction
  • At or in the place which
  • In whatever place
  • pronoun
  • Word you use to ask the location of something

leave

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US / li:v /

UK / liv /

A1
  • noun
  • Permission to do something
  • Vacation time; time off work
  • verb
  • To go away from; depart
  • To gift property to someone after you die
  • To allow something to continue happening
  • To let something remain unused or available
  • To let something or someone remain somewhere

hopefully

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US / ˈhəʊpfəli /

UK / ˈhopfəli /

A2
  • adverb
  • In a manner wishing something to happen

call

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US / kɔ:l /

UK / kɔl /

A1
  • noun
  • A order or request for action
  • The sound an animal makes, often when in danger
  • A correct prediction, e.g. a decision in a game
  • A short visit to a place or person
  • A decision by a referee or umpire in a game
  • verb
  • To make a request or order for action
  • (Animal) to make sounds to warn of danger
  • To announce the date of an election
  • To make a judgment about something
  • To give someone or something a name
  • To guess in advance the result or outcome
  • To visit a place or person for a short time
  • To phone someone
  • To speak, announce or shout in a loud voice

help

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US / help /

UK / hɛlp /

A1
  • noun
  • Something given to enable a person to do something
  • Assistance given to someone in need
  • Someone who makes something easier or possible
  • verb
  • To act to enable a person to do something; assist
  • To make something better or less difficult

minus

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US / ˈmaɪnəs /

UK / ˈmaɪnəs /

A2
  • adjecitve
  • Being close to the bottom of a letter grade
  • noun
  • Negative quality; disadvantage
  • Sign that indicates subtraction
  • preposition
  • Being less than zero in number
  • Used when subtracting one number from another
  • verb
  • To take one number away from another; subtract

and

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US / ənd /

UK / ənd, ən,ænd /

A1
  • conjunction
  • Used to refer to two or more things
  • Plus; in addition; on top of that
  • Used to introduce an action that follows another

over

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US / 'əʊvə(r) /

UK / ˈovɚ /

A1
  • adverb
  • Finished; no longer happening
  • Towards the ground/away from a standing position
  • noun
  • A set of six consecutive balls bowled in cricket
  • other
  • Finished
  • preposition
  • Showing the relation when a thing is above another
  • On or covering something
  • Above or across something
  • More than you need
  • From one side to the other
  • In the direction of; toward

blue

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US / blu: /

UK / blu /

A1
  • adjecitve
  • Color of the clear sky
  • Feeling sad or down

our

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US / 'aʊə(r) /

UK / aʊr /

A1
  • determiner
  • Belonging to us

pattern

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US / 'pætn /

UK / ˈpætən /

A2
  • noun
  • Model to follow in making or doing something
  • Colors or shapes which are repeated on objects
  • Regular repeated behavior
  • verb
  • To copy the way something else is made

deal

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US / di:l /

UK / dil /

A1
  • verb
  • To cope with something - usually troubles
  • To give (something bad e.g. news) to
  • To buy and sell illegal drugs
  • To give out (cards, etc.) to; distribute
  • To do business with someone or to sell products

same

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US / seɪm /

UK / sem /

A1
  • adjecitve
  • Being equal in quality and amount
  • Identical; not different
  • Not changing; constant

good

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US / ɡʊd /

UK / ɡʊd /

A1
  • adjecitve
  • Proper, appropriate or right
  • (Of an amount) enough; plenty
  • Excellent; high quality
  • Acting well or properly; of moral character
  • In a positive or happy emotional state
  • noun
  • Advantage or benefit

but

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US / bət /

UK / bʌt,bət /

A1
  • conjunction
  • Used before you say something different, opposite

apply

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US / ə'plaɪ /

UK / əˈplaɪ /

A1
  • verb
  • To spread a substance or liquid over a surface
  • To commit your time and effort to doing something
  • To make something useful in a certain situation
  • To be relevant for a situation
  • To ask formally for (job, permission etc.)

hat

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US / hæt /

UK / hæt /

A1
  • noun
  • Item of clothing worn on your head

color

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US / 'kʌlə(r) /

UK / ˈkʌlɚ /

A1
  • noun
  • Quality of things you can see, e.g. red, blue
  • Pink or red in your face, e.g. after being ill
  • The shade of someone's skin e.g. black, or yellow
  • verb
  • To change or affect someone's opinion
  • To make something colorful using colored pencils

not

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US / nɒt /

UK / nɑt /

A1
  • adverb
  • Word indicating the negative
  • (Used to form the negative of verbs)

equal

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US / 'i:kwəl /

UK / ˈikwəl /

A1
  • adjecitve
  • Same in shape, size, or number
  • verb
  • To add up or be the same as

stoke

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US / stəʊk /

UK / stok /

C1
  • verb
  • To add fuel or tend a fire to make it stronger

vector

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US / 'vektə(r) /

UK / ˈvɛktɚ /

B2
  • noun
  • Math a quantity that has both size and direction

video

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US / 'vɪdɪəʊ /

UK / ˈvɪdiˌo /

A2
  • adjecitve
  • Of a moving film, with sound
  • noun
  • Recording capturing action with sound
  • verb
  • To make a moving film of, with sound

dot

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US / dɒt /

UK / dɑt /

B1
  • noun
  • Small round mark
  • verb
  • To place small amounts/things in various places
  • To mark something by placing a small, round spot

give

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US / ɡɪv /

UK / ɡɪv /

A1
  • noun
  • Degree of flexibility in something, a material
  • verb
  • To hand over or present something to someone
  • To cause someone to have or experience something

now

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US / naʊ /

UK / naʊ /

A1
  • adverb
  • At the present time or moment
  • Right away or in the next possible moment
  • Used to refer to something that just happened
  • Time between the present and a moment in the past
  • conjunction
  • Resulting from or because of something

would

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US / wʊd /

UK / wʊd /

A1
  • other
  • Used to indicate something imagined
  • Used to ask or request something politely
  • Verb used to indicate possibility, request etc.
  • Used to indicate a preference
  • Used to indicate a reaction
  • Indicates something that occurred repeatedly
  • Used to indicate willingness or ability to do

want

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US / wɔnt /

UK / wɑnt, wɔnt /

A1
  • noun
  • Something you desire or wish for
  • State or condition of poverty
  • verb
  • To desire or wish for something; hope for a thing

can

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US / kæn /

UK / kən,kæn /

A1
  • noun
  • A metal container for storing food
  • other
  • Able to; have the skill to; have the time to
  • verb
  • To have permission to
  • To be possible to do
  • To preserve food by sealing it in a metal container

this

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US / ðɪs /

UK / ðɪs /

A1
  • adjecitve
  • Used to indicate something already been discussed
  • Used to indicate something happening around now
  • adverb
  • An indication by a physical motion or gesture
  • A specific amount of
  • determiner
  • Person, thing, or idea near you
  • pronoun
  • Person, thing that has already been discussed

how

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US / haʊ /

UK / haʊ /

A1
  • adjecitve
  • To what degree? in what amount?
  • adverb
  • (Used to explain the way to do something)
  • exclamation
  • For what purpose or reason?
  • (Used to ask someone or something's condition)

other

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US / 'ʌðə(r) /

UK / ˈʌðər /

A1
  • adjecitve
  • A thing different from that mentioned
  • determiner
  • (Something) else; not the first (one)
  • pronoun
  • Being the one and only thing of that category

the

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US / ðə /

UK / ðə /

A1
  • article
  • Used to refer to something already mentioned
  • Used to show there is only one of something
  • determiner
  • Used to describe all of a family

three

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US / θri: /

UK / θri /

A1
  • number
  • Number 3

path

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US / pɑ:θ /

UK / pæθ, pɑθ /

A2
  • noun
  • Method of living leading to a particular result
  • Track made with stones, by walking over the ground

into

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US / 'ɪntə /

UK / ˈɪntu /

A1
  • preposition
  • Moving or going inside something
  • In the same direction as; in the direction of
  • So as to become; taking the state of

kind

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US / kaɪnd /

UK / kaɪnd /

A1
  • adjecitve
  • In a caring and helpful manner
  • friendly and considerate
  • noun
  • One type of thing

play

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US / pleɪ /

UK / pleɪ /

A1
  • noun
  • Loose, free movement between two objects
  • Single action in a sports game
  • Performance in which people act in a theater
  • verb
  • To act as a character in a movie, or on stage
  • To perform music on an instrument
  • To use a device, e.g. a video game
  • To do something for enjoyment and fun
  • To do or perform a game or sport

make

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US / meɪk /

UK / mek /

A1
  • noun
  • Product made by a particular company; brand
  • verb
  • To arrange or prepare something e.g. dinner
  • To get to (a place) on time
  • To gain the status of (teacher, lawyer, etc.)
  • To create something by putting things together
  • To earn a certain amount of money at a job
  • To cause or force a person to do something
  • To cause something to happen or be formed

attempt

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US / ə'tempt /

UK / əˈtɛmpt /

A2
  • noun
  • Effort made to try to do or accomplish something
  • verb
  • To try to do something challenging or difficult

let

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US / let /

UK / lɛt /

A1
  • verb
  • To allow someone to do something
  • Introducing a suggestion to do something together
  • To rent a house, etc. to others

surface

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US / 'sɜ:fɪs /

UK / ˈsɚfəs /

A2
  • adjecitve
  • Of the top layer; not deep or meaningful
  • noun
  • Top layer of the ground or of water
  • Nature or emotions that show, but may not be true
  • Outside or upper layer of something
  • verb
  • To give (road) a top layer
  • To appear after being hidden, unseen, or unknown
  • To come to the top of something; emerge
  • To come to the top level of water, mud, etc.

you

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US / jʊ /

UK / ju /

A1
  • pronoun
  • Person someone is speaking or writing to
  • Person or people in general

little

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US / 'lɪtl /

UK / ˈlɪtl /

A1
  • adjecitve
  • Being short in length of time
  • Not great in size, amount, or degree; small
  • Being very young or younger than others
  • adverb
  • By an amount or degree that is not very much
  • Not often; rarely
  • noun
  • A person's name

one

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US / wʌn /

UK / wʌn /

A1
  • pronoun
  • You; we; people in general; someone
  • The thing or person previously referred to
  • adjecitve
  • Used to refer to people generally
  • Being a strong example of (something mentioned)
  • number
  • Number 1

entire

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US / ɪn'taɪə(r) /

UK / ɛnˈtaɪr /

A2
  • adjecitve
  • Complete or full; with no part left out; whole

order

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US / 'ɔ:də(r) /

UK / ˈɔrdɚ /

A1
  • noun
  • Religious or social group, as of nuns
  • Instructions to others to do what you want
  • State in which things are neat and organized
  • What you ask to buy, e.g. food in restaurant
  • Social or political control of an area
  • verb
  • To put things in a sequence, e.g. a, b, c
  • To give instructions to others to do what you want
  • To ask for something you want to buy, e.g. a meal

assumption

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US / ə'sʌmpʃn /

UK / əˈsʌmpʃən /

A2
  • noun
  • Something you believe to be so, but aren't sure of

half

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US / hɑ:f /

UK / hæf, hɑf /

A1
  • adjecitve
  • Not complete
  • adverb
  • Partly; somewhat
  • noun
  • One of two equal periods of game time in a sport
  • The part you get when one is divided into two;

able

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US / 'eɪbl /

UK / ˈebəl /

A2
  • adjecitve
  • Skilled; good at something
  • Having the ability to do something

checkerboard

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US / ˈtʃekəbɔ:d /

UK / ˈtʃekərbɔ:rd /

A1
  • noun
  • Board having squares of two alternating colors

case

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US / keɪs /

UK / kes /

A1
  • noun
  • Container used to carry things, e.g. clothes
  • Set of facts about a crime
  • Set of facts about an event
  • Example or instance of something
  • verb
  • To find out about a place, as in planning to steal

second

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US / 'sekənd /

UK / ˈsɛkənd /

A1
  • adjecitve
  • 2nd; the one following first
  • noun
  • Unit of time; one sixtieth of a minute
  • Short period of time that you wait for someone
  • (Do) soon

care

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US / keə(r) /

UK / kɛr /

A1
  • noun
  • Treatment you receive for your health or safety
  • Something that a person worries about
  • verb
  • To feel interest, concern, or worry
  • To look after someone or something

convincing

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US / kənˈvɪnsɪŋ /

UK / kənˈvɪnsɪŋ /

A2
  • verb
  • To persuade someone, or make them feel sure

respect

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US / rɪ'spekt /

UK / rɪˈspɛkt /

A2
  • noun
  • Behavior that follows the established rules
  • Admiration for someone
  • Regard or admiration for someone or something
  • verb
  • To follow the established rules
  • To think very highly of another person

then

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US / ðen /

UK / ðɛn /

A1
  • adverb
  • A time that is not now
  • Following another thing in time or arrangement
  • Following on reasonably from what was just stated
  • At that time, not now

different

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US / ˈdɪfrənt /

UK / ˈdɪfrənt /

A1
  • adjecitve
  • Not of the same kind; unlike other things

each

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US / i:tʃ /

UK / itʃ /

A1
  • determiner
  • Every one of two or more things
  • pronoun
  • Per one person or thing

boundary

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US / 'baʊndrɪ /

UK / ˈbaʊndəri, -dri /

B1
  • noun
  • A division or border to separate two areas

essentially

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US / ɪˈsenʃəli /

UK / ɪˈsenʃəli /

A2
  • adverb
  • Basically; (said when stating the basic facts)

find

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US / faɪnd /

UK / faɪnd /

A1
  • noun
  • Object discovered by chance
  • Things that have been discovered from searching
  • verb
  • To become aware of something that is happening
  • To discover or meet by chance
  • To decide if someone is guilty of a crime, or not
  • To learn through study, research, or effort
  • To discover something by looking for it

that

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US / ðæt /

UK / ðæt, ðət /

A1
  • adjecitve
  • The person or thing being talked about
  • Used to identify something both the speakers know
  • adverb
  • Used to reinforce adjectives and adverbs
  • To such a degree
  • conjunction
  • Used to connect a noun clause
  • Used to introduce an adverbial clause
  • Used to introduce a noun clause
  • determiner
  • Used as a noun to refer to something
  • pronoun
  • Referring an object far away
  • Used to refer to the relative pronoun ‘which’

tell

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US / tel /

UK / tɛl /

A1
  • verb
  • To (strongly) advise a person to do something
  • To be able to guess
  • To say or communicate information to someone
  • To inform, but without speaking; indicate
  • To have a negative effect on someone
  • To order someone to do something; command
  • To recognize something based on certain signs
  • To report something to an authority
  • To explain something or teach someone something

like

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US / laɪk /

UK / laɪk /

A1
  • adjecitve
  • Being equal in status or kind to something else
  • Similar; the same
  • adverb
  • For example
  • preposition
  • Used as a filler; used to ad emphasis
  • In a way similar to the way something is done
  • Such as; for example
  • verb
  • To want to
  • To find something pleasing; to prefer something

about

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US / ə'baʊt /

UK / əˈbaʊt /

A1
  • adverb
  • To be ready to; going to
  • Nearly; approximately; roughly
  • Near; next to; close to
  • In every direction; in the area of
  • preposition
  • Concerning or of (a topic)

curl

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US / kɜ:l /

UK / kɚrl /

B2
  • noun
  • Something that has the shape of a curve or ring
  • verb
  • To form something into a round or curved shape
  • To become curved around the edges

get

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US / ɡet /

UK / ɡɛt /

A1
  • verb
  • To become affected by illness or disease
  • To begin to feel or understand an emotion or idea
  • To send or bring someone or something home, etc.
  • To go somewhere to obtain something
  • To (cause to) do a particular thing
  • To obtain, receive or be given something
  • To currently have
  • To prepare for use
  • To understand something being said or read

what

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US / wɒt /

UK / hwɑt, hwʌt, wɔt, wʌt,hwət, wət /

A1
  • pronoun
  • Thing or things that; the same kind as
  • Used to emphasize something you are about to say
  • adjecitve
  • Used to refer to a particular amount, or number
  • Used to place emphasis on a particular quality
  • adverb
  • To a certain degree
  • determiner
  • Question words used for asking for information
  • exclamation
  • Used to express excitement, shock, or surprise
  • Used to ask people to repeat something they said

operator

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US / ˈɒpəreɪtə(r) /

UK / ˈɑ:pəreɪtə(r) /

A2
  • noun
  • Someone who operates or controls a machine
  • Person whose job it is to connect phone calls

right

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US / raɪt /

UK / raɪt /

A1
  • adjecitve
  • Correct or true
  • The south side of you looking at the rising sun
  • adverb
  • Exactly in a place or time
  • Being appropriate for a particular event
  • In a straight or direct manner
  • noun
  • Something you are legally or morally allowed to do
  • Ideal of what is just and good
  • Political view based on conservatism
  • verb
  • To fix something previously wrong or unjust
  • To make upright, as a boat that has turned over

with

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US / wɪð /

UK / wɪð, wɪθ /

A1
  • preposition
  • As an employee of a certain company
  • Against something or someone
  • By using something as a means
  • Due to; because of
  • Including
  • Possessing as a feature
  • Related to
  • (Of the manner/attitude by which you do something)
  • Used to express that people or things are together

blank

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US / blæŋk /

UK / blæŋk /

A2
  • verb
  • To make empty
  • To cause a memory to be erased
  • adjecitve
  • Showing no emotion or understanding
  • Being empty; not filled out; with no writing on it
  • noun
  • Empty space on a form which you fill in

see

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US / si: /

UK / si /

A1
  • verb
  • To go on a date with someone
  • To predict whether something is, or will be true
  • To understand what someone says
  • To use your eyes to look at something
  • To visit a place often for sightseeing or pleasure
  • To visit or meet with someone
  • To find out by waiting or looking
  • To watch (a game, movie or TV show)

theorem

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US / ˈθɪərəm /

UK / ˈθiərəm, ˈθɪrəm /

B2
  • noun
  • A statement that can be proved

could

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US / kəd /

UK / kʊd /

A1
  • other
  • Past form of 'can' to mean have ability
  • Used to ask for something politely, or to offer

next

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US / nekst /

UK / nɛkst /

A1
  • adverb
  • The one that follows another in time or order
  • At a time or place directly after the present one
  • noun
  • The one that follows after
  • adjecitve
  • Immediately after the previous one
  • First occurrence after the present one

focus

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US / 'fəʊkəs /

UK / ˈfokəs /

A2
  • noun
  • Main purpose or center of interest or activity
  • verb
  • To see clearly by adjusting your eyes or a camera
  • To keep your attention, concentration on one thing

have

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US / həv /

UK / hæv /

A1
  • verb
  • To drink, smoke, eat or use something
  • To experience the effects of something
  • To organize an event, or join in an activity
  • To own, possess, or hold something
  • To cause to happen or produce a particular effect

mapping

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US / ˈmæpɪŋ /

UK / ˈmæpɪŋ /

A2
  • verb
  • To draw a diagram to show the geography of an area
  • To find points of similarity between things

figure

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US / ˈfiɡə /

UK / ˈfɪɡjɚ /

A1
  • noun
  • Your body shape
  • Numbers in a calculation
  • Doll-like thing meant to represent a person
  • Picture or diagram giving information in a text
  • Person who is very important or famous
  • Shape of a person seen indistinctly or in outline
  • Amount that is expressed in numbers
  • verb
  • To appear in a game, play or event
  • To calculate how much something will cost
  • To understand after thinking; work out

simple

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US / 'sɪmpl /

UK / ˈsɪmpəl /

A1
  • adjecitve
  • Not hard to understand or do; not complex
  • Having only a few parts; not complex
  • Being ordinary or normal
  • Being plain; without a lot added for style

bit

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US / bɪt /

UK / bɪt /

A1
  • noun
  • Device put in a horse's mouth to control it
  • Small piece of something
  • verb
  • (E.g. of fish) to take bait and be caught

take

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US / teɪk /

UK / tek /

A1
  • noun
  • Filming of a single scene for TV or the movies
  • verb
  • To accept something that is offered
  • To use a method of traveling, e.g. a bus or train
  • To use a camera to make (a photo)
  • To pick up something and go away with it
  • To study a subject in school
  • To attempt to pass (an exam)
  • To require a certain amount of time, money, space

they

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US / ðeɪ /

UK / ðe /

A1
  • pronoun
  • Two or more people, animals, or things

way

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US / weɪ /

UK / we /

A1
  • adverb
  • Used to emphasize the amount of something
  • noun
  • Particular aspect of something being thought about
  • One of various choices that could be made
  • Particular existing condition or state
  • (Often long) distance or length of time
  • How something is done
  • Route to go to a place
  • Small street
  • Usual behavior, emotions or habits of someone

derivative

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US / dɪˈrɪvətɪv /

UK / dɪˈrɪvətɪv /

B1
  • adjecitve
  • Not from original ideas; from somewhere else

pretty

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US / 'prɪtɪ /

UK / ˈprɪti /

A1
  • adjecitve
  • Being attractive to the eye in a simple way
  • adverb
  • Very; to a great degree

prove

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US / pru:v /

UK / pruv /

A2
  • verb
  • To demonstrate truth by providing evidence

field

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US / fi:ld /

UK / fild /

A1
  • noun
  • Area of study, such as physics or biology
  • Piece of land used to grow crops/raise animals
  • Open area of land, especially without buildings
  • Grassed area where you play some sports
  • verb
  • To respond to something or answer a question
  • To catch or stop a ball during a game

statement

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US / ˈsteɪtmənt /

UK / ˈstetmənt /

A2
  • noun
  • Record of activity in a bank account over time
  • Act or process of saying something formally

just

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US / dʒʌst /

UK / dʒʌst /

A1
  • adjecitve
  • Proper or appropriate; as is deserved
  • Doing or being what is right or fair
  • Legally correct; having enough evidence
  • adverb
  • Almost; nearly
  • Exactly at the time of
  • Exactly (the same as, like)

short

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US / ʃɔ:t /

UK / ʃɔrt /

A1
  • adverb
  • Less than what is needed
  • Suddenly stopping what you are saying or doing
  • verb
  • (Of electric circuit) to spark because faulty
  • adjecitve
  • Measuring a small distance
  • Happening for a small amount of time
  • Unfriendly or rude

express

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US / ɪk'spres /

UK / ɪkˈsprɛs /

A1
  • adjecitve
  • Sent by a fast mail service
  • Traveling at high speed
  • noun
  • Fast train or bus
  • verb
  • To send something by fast mail
  • To state or show what you think or how you feel

column

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US / ˈkɒləm /

UK / ˈkɑ:ləm /

B1
  • noun
  • Regular series of articles in a newspaper/magazine
  • Strong vertical stone pillar that holds up a roof
  • Vertical line in a table or chart

operation

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US / ˌɒpəˈreɪʃn /

UK / ˌɑpəˈreʃən /

A2
  • noun
  • Business or company, or its activities
  • Functioning; use
  • Medical procedure involving surgery

green

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US / gri:n /

UK / ɡrin /

A1
  • adjecitve
  • Color of young leaves
  • Covered with vegetation, e.g. grass
  • Concerned about protecting the environment
  • The color of young leaves
  • Lacking in experience/knowledge due to being young
  • Being not fully grown or ripe, as in fruit
  • noun
  • Large grassy area
  • used as a part of a name of a place (UK)

determine

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US / dɪ'tɜ:mɪn /

UK / dɪˈtɚmɪn /

A2
  • verb
  • To control exactly how something will be or act
  • To establish the facts about; discover

partial

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US / ˈpɑ:ʃl /

UK / ˈpɑ:rʃl /

B1
  • adjecitve
  • Giving better treatment to one person than another
  • Not complete; being only part of something

for

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US / fə(r) /

UK / fɔr,fə /

A1
  • preposition
  • Used to show the purpose, or need of something

well

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US / wel /

UK / wɛl /

A1
  • adjecitve
  • Being in a fortunate, good condition or state
  • Healthy and in good condition
  • adverb
  • In an acceptable, proper, or successful manner
  • In a complete, total, or full manner
  • A great and considerable degree
  • In a successful way, in a good or satisfactory way
  • exclamation
  • Used to begin or continue a particular statement
  • Used to show you want to start saying something
  • Used to express that you are waiting for a reply
  • You say this to change the topic of a conversation
  • Used to indicate you are making a final remark
  • noun
  • Long deep hole in the ground to get water
  • verb
  • To cause your eyes to fill with tears

time

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US / taɪm /

UK / taɪm /

A1
  • noun
  • Speed at which music is played; tempo
  • Point as shown on a clock, e.g. 3 p.m
  • Number of hours, minutes needed to do something
  • Occasion when something happens
  • Period or occasion that something occurred
  • Period in history or the past
  • Something measured in minutes, hours, days, etc.
  • How long an event takes; duration
  • verb
  • To check speed at which music is performed
  • To choose a specific moment to do something
  • To measure how long an event takes, e.g. a race
  • To schedule something to occur at a specific moment

why

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US / waɪ /

UK / hwaɪ, waɪ /

A1
  • adverb
  • Question word to ask for a reason
  • exclamation
  • (Used to express surprise or approval)

special

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US / 'speʃl /

UK / ˈspɛʃəl /

A1
  • adjecitve
  • Being additional or extra
  • Different, better or greater than normal
  • noun
  • Heavily discounted sales item

start

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US / stɑ:t /

UK / stɑrt /

A1
  • noun
  • First time or place that a thing exists; beginning
  • First opportunity to achieve something, e.g. a job
  • Sudden action or movement because you are scared
  • other
  • Beginning of something in place or time
  • verb
  • To do, be or happen for the first time; begin
  • To turn something on
  • Subtitles section
  • In this video, I will attempt to prove a special case version of Stokes' Theorem, or essentially, Stokes' Theorem for a special case.
  • And I'm doing this because the proof will be a little bit simpler.
  • But at the same time, it's pretty convincing.
  • And the special case we're going to assume is that the surface we're dealing with is a function of x and y.
  • So if you give me any particular x and y, it only determines one point on that surface.
  • So a surface like this would be the case.
  • So it's kind of a mapping of this region in the xy plane into three dimensions.
  • So for any xy, we can figure out the height.
  • So essentially, z is going to be a function of x and y.
  • And we can get a point on the surface.
  • So this proof would not apply to a surface that's like a sphere or something like that, where any point on the xy plane could actually determine two points on our surface.
  • But this is a pretty good start.
  • The other thing that we're going to assume, we're going to assume that z, which is essentially a function of x and y, that this function of x and y has continuous second-order derivatives.
  • And the reason why I'm going to make that assumption is it's going to help us in our proof later on.
  • It's going to allow us to say that the partial of our surface or the partial of z with respect to x, and then taking the derivative of that with respect to y, is going to be the same as the partial of z with respect to y,
  • And then taking the derivative of that with respect to x.
  • And in order to be able to make this statement, we have to assume that z, or this z right over here, z is a function of x and y, has continuous second-order derivatives.
  • And over here, we've just written our vector field f that we're going to deal with when we're trying to play with Stokes' Theorem.
  • And we'll assume that it has continuous first-order derivatives.
  • Now with that out of the way, let's think about what Stokes' Theorem tells us.
  • And then we'll think about, for this particular case, how we can write it out.
  • And then hopefully, we will see that the two things are equal.
  • So let me write it out.
  • So Stokes' Theorem tells us that f dot dr over some path, and the path that we care about is essentially this path right over here.
  • I'll do it in blue.
  • It's this path right over here.
  • This is the boundary.
  • This is the boundary of our surface.
  • So this is c right over here.
  • Stokes' Theorem tells us that this should be the same thing.
  • This should be equivalent to the surface integral over our surface of curl of f dot ds, dot dotted with the surface itself.
  • And so in this video, I want to focus, or probably this and the next video, I want to focus on the second half.
  • I want to focus this.
  • I want to see how we can express this, given the assumptions we've made.
  • And then after that, we're going to see how we can express this, given the same assumptions.
  • And then hopefully, we'll find that we get them to be equal to each other.
  • So let's just start figuring out what the curl of f is equal to.
  • So the curl of f is equal to, you could view it as the del operator crossed with our vector field f, which is equal to, we can write our components.
  • So i, I'll do them in different colors, i, j, and k components.
  • And then I need to write my del operator, or my partial operators, I guess I should call them.
  • So the partial with respect to x, the partial with respect to y, partial with respect to z.
  • And then I have to write the i, j, and k components of my vector field f.
  • And I will do that in green.
  • Well, I'll do it in blue.
  • And so we have p, which is a function of x, y, z.
  • Q, which is a function of x, y, and z.
  • And then r, which is a function of x, y, and z.
  • And so this is going to evaluate as, it's going to be i times, so blank out that column, that row.
  • It's going to be the partial of r with respect to y.
  • Minus the partial of q with respect to z.
  • And then checkerboard pattern, minus j.
  • Make that hat a little bit better.
  • Minus j.
  • And then times the partial of r with respect to x.
  • Minus the partial of p with respect to z.
  • Minus the partial of p with respect to z.
  • And then finally, plus k.
  • And that's going to be times the partial of q with respect to x.
  • Minus the partial of p with respect to y.
  • So we figured out the curl of f.
  • And I'll leave you there.
  • I'm trying to make shorter videos now.
  • And so in the next video, we have to figure out how to express ds.
  • And then we'll just evaluate this entire operation.
  • We'll take the dot product.