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Stokes' theorem proof part 5 | Multivariable Calculus | Khan Academy

Video vocabulary

essentially

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US / ɪˈsenʃəli /

UK / ɪˈsenʃəli /

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

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

notation

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US / nəʊ'teɪʃn /

UK / noˈteʃən /

C2
  • noun
  • Set of written signs are used in music or math

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

boundary

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US / 'baʊndrɪ /

UK / ˈbaʊndəri, -dri /

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

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

careless

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US / 'keələs /

UK / ˈkɛrlɪs /

A2
  • adjecitve
  • Not being careful

down

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US / daʊn /

UK / daʊn /

A1
  • adjecitve
  • Being sad, sick, or unhappy
  • Not operating, working, or functioning as expected
  • adverb
  • In a sitting or lying position
  • Towards or on the ground, floor, or bottom
  • Written or copied onto a piece of paper or book
  • noun
  • A period of play in American Football
  • The soft, smooth hairs that cover a bird's body
  • preposition
  • Along something's path, course, or direction
  • Going from a higher position to a lower position
  • verb
  • To eat or drink very quickly and without stopping

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

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)

again

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US / ə'ɡen /

UK / əˈɡɛn /

A1
  • adverb
  • One more time, once more

different

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US / ˈdɪfrənt /

UK / ˈdɪfrənt /

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

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’

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

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

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

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

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

correspond

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US / ˌkɒrəˈspɒnd /

UK / ˌkɔ:rəˈspɑ:nd /

B1
  • verb
  • To have similarity or equality with something
  • To write letters or emails to someone

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

from

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US / frəm /

UK / frʌm, frɑm,frəm /

A1
  • preposition
  • Starting at a particular place, time or level

form

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US / fɔ:m /

UK / fɔrm /

A1
  • noun
  • Sports team or person's current winning record
  • Document you complete when making an application
  • Visible shape or style; type; kind
  • verb
  • To organize something such as a club or group
  • To develop; to come into a shape or substance
  • To make something into a particular shape

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

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

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

rearrange

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US / ˌri:ə'reɪndʒ /

UK / ˌriəˈrendʒ /

B1
  • verb
  • To rethink a plan, event for a time and place
  • To put into a new order for something to happen

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

side

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US / saɪd /

UK / saɪd /

A1
  • adjecitve
  • In addition to the central or most important thing
  • Next to, or very close to, the right or left of
  • noun
  • Position or opinion that is opposite to another
  • Right or left part of a person's body
  • Feature of someone's personality
  • Straight edge of an object
  • One of two or more groups fighting each other
  • verb
  • To decide to agree with one point, not the other

actually

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US / ˈæktʃuəli /

UK / ˈæktʃuəli /

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

integral

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US / ˈɪntɪgrəl /

UK / ˈɪntɪɡrəl, ɪnˈtɛɡrəl /

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

use

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US / ju:s /

UK / ju:s /

A1
  • noun
  • Ability to employ something
  • Need to enable it to work
  • Permission to do something with something
  • Act of spending money
  • Operation of a machine for its function
  • verb
  • To have; to eat, drink, consume etc.
  • To need to operate or work
  • To operate a machine, e.g. a computer
  • To treat a person badly
  • To do something with, for a task or purpose

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

here

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US / hɪə(r) /

UK / hɪr /

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

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

afraid

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US / əˈfreɪd /

UK / əˈfred /

A1
  • adjecitve
  • Worried that something bad will happen; scared

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

once

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US / wʌns /

UK / wʌns /

A1
  • adverb
  • One time; one instance
  • At a time in the past
  • conjunction
  • When; as soon as

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

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

also

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US / 'ɔ:lsəʊ /

UK / ˈɔlso /

A1
  • adverb
  • In addition; too; in a similar way

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)

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

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

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

material

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US / məˈtɪəriəl /

UK / məˈtɪriəl /

A2
  • adjecitve
  • Relevant; (of evidence) important or significant
  • Belonging to the world of physical things
  • noun
  • Cloth; fabric
  • Supplies or data needed to do a certain thing
  • Substance from which a thing is made of

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

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.

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

separately

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US / ˈseprətli /

UK / ˈsɛpərɪtlɪ /

A2
  • adverb
  • Not with another person or thing

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

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

change

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US / tʃeɪndʒ /

UK / tʃendʒ /

A1
  • noun
  • Exchange of one set of clothes for another
  • Money in the form of coins instead of paper
  • Money returned after giving too much
  • Act of making or becoming something else
  • verb
  • To exchange one set of clothes for another
  • To exchange one kind of money for another
  • To replace something with another thing
  • To make or become something else
  • To go from one train, bus, etc. and go to another

due

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US / dju: /

UK / du, dju /

A1
  • adjecitve
  • When something is required or expected
  • Owed to a person as a right
  • noun
  • Something you deserve; something you can expect

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

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

all

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US / ɔ:l /

UK / ɔl /

A1
  • adverb
  • Completely; totally
  • determiner
  • Being every one of something
  • pronoun
  • 100% of something; the complete amount

its

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US / ɪts /

UK / ɪts /

A1
  • adjecitve
  • Referring to or belonging to ‘it’
  • determiner
  • Of something; belonging or related to something
  • other
  • Possessive form of “it”

fast

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US / fɑ:st /

UK / fæst /

A1
  • adjecitve
  • In a way that is difficult to move or change
  • Moving or able to move quickly
  • Showing time that is later than the actual time
  • adverb
  • With a great deal of speed
  • noun
  • Period of time when a person chooses not to eat
  • verb
  • To refuse food for a long time

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

derivative

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US / dɪˈrɪvətɪv /

UK / dɪˈrɪvətɪv /

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

multiply

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US / ˈmʌltɪplaɪ /

UK / ˈmʌltəˌplaɪ /

B1
  • verb
  • To increase in number by reproducing/having babies
  • To increase something by a large quantity
  • To have increased by a large quantity
  • To calculate two numbers e.g. 2 x 4

rewrite

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US / ˌri:'raɪt /

UK / riˈraɪt /

A2
  • verb
  • To write again to improve, or correct errors

prove

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US / pru:v /

UK / pruv /

A2
  • verb
  • To demonstrate truth by providing evidence

throw

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US / θrəʊ /

UK / θroʊ /

A1
  • noun
  • Arm movement to make a thing fly through the air
  • Loose cloth or blanket (usually over a chair)
  • Forcibly putting someone on the ground
  • verb
  • To use your arm to make something fly in the air
  • To move part of your body suddenly and forcefully
  • To confuse or upset someone
  • (E.g. judo) to forcibly put someone on the ground
  • To put something somewhere suddenly and roughly

between

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US / bɪˈtwi:n /

UK / bɪˈtwin /

A1
  • preposition
  • From one person, thing, or place, to another
  • In the space that separates two objects
  • Among two or more people who share something

everything

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US / 'evrɪθɪŋ /

UK / ˈɛvriˌθɪŋ /

A1
  • pronoun
  • All of the things mentioned

original

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US / ə'rɪdʒənl /

UK / əˈrɪdʒənəl /

A2
  • adjecitve
  • Being first made, thought or performed; fresh
  • Being unique or thinking differently from others

they

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US / ðeɪ /

UK / ðe /

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

add

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US / æd /

UK / æd /

A1
  • verb
  • To include as well
  • To put two things or numbers together, e.g. 2+2

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

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)

line

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US / laɪn /

UK / laɪn /

A1
  • noun
  • Border that divides two places or regions
  • Long crease in the skin, often from age
  • Mark that is long, straight and very thin
  • Row of words, letters or numbers that are written
  • Telephone connection
  • People waiting one behind the other; queue
  • Long thin wire, e.g. for electrical signal
  • verb
  • To put material along the inside of something

talk

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US / tɔ:k /

UK / tɔk /

A1
  • noun
  • Style of speaking
  • Discussion between two countries
  • Giving information in front of people; lecture
  • Saying things or ideas to someone with words
  • verb
  • To make a formal speech about something
  • To say things or ideas to someone with words

conceptualize

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US / kənˈseptʃuəlaɪz /

UK / kənˈsɛptʃuəˌlaɪz /

C1
  • verb
  • To have the ability to see concepts or ideas

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

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

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

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

for

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US / fə(r) /

UK / fɔr,fə /

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

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

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

algebraic

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US / ˌældʒɪ'breɪɪk /

UK / ˌældʒəˈbreɪk /

C1
  • adjecitve
  • Concerning algebra

foundational

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US / faʊn'deɪʃənəl /

UK / faʊn'deɪʃənəl /

C1
  • adjecitve
  • Being the base or important starting point

those

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US / ðəʊz /

UK / ðoz /

A1
  • determiner
  • The people or things we are talking about
  • pronoun
  • The objects far from the speaker

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

slightly

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US / ˈslaɪtli /

UK / ˈslaɪtli /

B1
  • adverb
  • Only a little

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

break

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US / breɪk /

UK / brek /

A1
  • noun
  • Time you stop an activity before continuing
  • End of a relationship
  • An escape from a place such as a jail
  • (Lucky) advantage or benefit from something
  • Find a solution or answer to a problem or crime
  • Change in the weather
  • verb
  • To create a new record e.g. running the 100m dash
  • To form a crack in something
  • To burst or become damaged
  • To end a relationship
  • To escape from a place such as a jail
  • To train a wild animal e.g. a horse
  • To separate into pieces by force, or by dropping
  • To act against a law, rule, or promise
  • To become known suddenly, as in the news
  • To not do what you promised e.g. not keep promises
  • To solve a crime, or answer a problem
  • To stop functioning properly
  • To suddenly start, open or commence
  • To change in pitch suddenly, as in a voice
  • To fall on the shore, as in waves
  • To begin or change suddenly; the dawn or weather
  • To defeat by causing the person to lose their will

vector

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US / 'vektə(r) /

UK / ˈvɛktɚ /

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

but

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US / bət /

UK / bʌt,bət /

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

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

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.)

same

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US / seɪm /

UK / sem /

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

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

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

stoke

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US / stəʊk /

UK / stok /

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

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

whole

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US / həul /

UK / hol /

A1
  • adjecitve
  • Complete or full; all of
  • Being in a good condition
  • noun
  • Complete amount or sum; total

our

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US / 'aʊə(r) /

UK / aʊr /

A1
  • determiner
  • Belonging to us

not

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US / nɒt /

UK / nɑt /

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

manipulation

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US / məˌnɪpjʊ'leɪʃn /

UK / məˌnɪpjʊˈleʃən /

B1
  • noun
  • Act of controlling to gain an advantage
  • Act of moving something using your hands

really

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US / ˈri:əli /

UK / ˈriəˌli, ˈrili /

A1
  • adverb
  • Used to stress something that is said; definitely
  • In a true or accurate manner
  • Very

business

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US / ˈbɪznəs /

UK / ˈbɪznɪs /

A1
  • noun
  • A company formed for making profit
  • Matter that has to be dealt with; task; situation
  • Activity of working to make a profit

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

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

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

copy

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US / ˈkɒpi /

UK / ˈkɑ:pi /

A1
  • noun
  • A counter for items that are the same
  • Fake product pretending to be an original
  • An imitation of an original
  • A duplicate of a paper printed in a photocopier
  • Written text for printing, use in ads, etc.
  • verb
  • To make something look the same as the original
  • To act in the exact same way as someone else
  • To read and then write down what you read

theorem

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US / ˈθɪərəm /

UK / ˈθiərəm, ˈθɪrəm /

B2
  • noun
  • A statement that can be proved

remember

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US / rɪ'membə(r) /

UK / rɪˈmɛmbɚ /

A1
  • verb
  • To give someone a gift, e.g. birthday, wedding
  • To hold a thought in your mind to not forget it
  • To bring a previous image or idea to your mind

end

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US / end /

UK / ɛnd /

A1
  • noun
  • The point where a process or activity stops
  • A point that marks the limit of something; finish
  • Goal or desired result
  • The tip, edge or limit of something
  • verb
  • To be the last part of something

clear

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US / klɪə(r) /

UK / klɪr /

A1
  • adjecitve
  • Empty; without anything in it or on it
  • Being certain and without doubt
  • Being without any streaks, spots, or cracks
  • Easy to understand; well-explained; obvious
  • adverb
  • Completely; entirely; through the whole of
  • Away from; free from
  • verb
  • To recover from or remove a problem
  • To get rid of the entire contents of
  • To give permission for something to happen
  • To make a place or surface empty by removing
  • To prove that someone is not guilty or to blame

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)

part

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US / pɑ:t /

UK / pɑ:rt /

A1
  • adjecitve
  • To some degree; half; not fully
  • noun
  • Division of a book
  • Ratio of something, e.g. 3 of gin, 1 of tonic
  • A line in a person's hair, made with a comb
  • Character an actor plays on stage, movie, or TV
  • Some, but not all of a specific thing
  • Role in causing something to happen
  • verb
  • To make a line in a person's hair, by using a comb
  • To move two things apart creating a space
  • To leave someone, or to break off a relationship

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

before

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US / bɪˈfɔ:(r) /

UK / bɪˈfɔr,-ˈfor /

A1
  • adverb
  • At a time earlier than the present; previously
  • conjunction
  • In advance of something happening
  • preposition
  • In front of someone or something
  • Subtitles section
  • Now that we have a parameterization for the boundary of our surface right up here, let's think a little bit about what the line integral-- and this was the left side of our original Stokes' theorem statement-- what the line integral over the path C of F, our original vector field F, dot dr is going to be.
  • And now, once again, this R we're talking about the path of this boundary right over here, C. Not C1 that we did to kind of get some of our foundational material going.
  • So F dot dr-- well, we remember what F is.
  • F was all the way up here.
  • Its i, j, and k components were just P, Q, and R.
  • So that's kind of easy to remember.
  • But let's think about what dr is equal to.
  • And we're going to have to break out a little bit of our three-dimensional or our multivariable chain rule right over here.
  • So dr is the same thing as dr dt times dt.
  • So we really just have to figure out what the derivative of R with respect to t.
  • And this one we're going to have to break out a little bit of the chain rule.
  • So let me write this down.
  • So dr dt is going to be equal to the derivative of x with respect to t times i plus the derivative of y with respect to t-- I'm just taking the derivative with respect to t-- j.
  • Because z is a function of x, which is a function of t, and z is also a function of y, which is a function of t, we're going to have to break out our multivariable chain rule.
  • So if we want to take the derivative of z with respect to t-- I'll do it separately here, and then I'll write it down down here.
  • The derivative of z with respect to t-- the way I conceptualize is, what's all the different ways that z can change from a change in t?
  • Well, it could change because x is changing due to a change in t.
  • So z could change due to x, the partial of z with respect to x, when x changes with respect to t.
  • But then that's not the only way that z can change.
  • We have to add to that how z can change with respect to y, partial of z with respect to y times how fast or how y is changing with respect to t.
  • And this is just our multivariable chain rule.
  • And so this is dz dt.
  • So I'll just write it right over here.
  • And I'll use slightly different notation that's consistent with what we were doing before, and it'll help make things a little bit clearer.
  • So it's going to be the partial of z with respect to x, dx dt, plus-- actually, let me write it this way-- plus the partial of z with respect to y dy dt.
  • And then we're going to multiply everything times our k.
  • So with this out of the way, if we wanted what dr is, dr is just going to be this whole thing times dt.
  • So let's do that.
  • So now we can rewrite our line integral right over here.
  • And we're going to now go into the t domain.
  • And so t is going to go between a and b.
  • And F dot dr-- remember, F's components were just the functions P, Q, and R. And each of those were functions of x, y, and z.
  • And z is a function of x and y, so we'll have to think about all of that in a little bit.
  • Use a little more of our multivariable chain rule.
  • But when we take the dot products, we're just going to take the corresponding components and multiply them.
  • So it is going to be-- actually, let me just copy and paste it.
  • Let me rewrite it down here.
  • Our vector field F-- and I'm going to write it a little bit shorter.
  • Our vector field F is P times i plus Q times j plus R times k.
  • So when we take our dot product of F dot dr, we're essentially taking the dot product of this and this.
  • And we have to throw a dt at the end.
  • So we're going to get P times dx dt plus Q times dy dt plus R times all of this business over here, which is the partial of z with respect to x dx dt plus the partial of z with respect to y dy dt.
  • And then we have to multiply all of this times dt.
  • We can't forget that part right over there.
  • So we're going to multiply all of this right here times dt.
  • Now, I'm going to leave you there in this video just because I'm afraid of making careless mistakes.
  • What we're going to do now is we're going to rearrange this whole thing, recognize that this is the same thing as this right over here.
  • Then that gives it a form that we can apply Green's theorem using this boundary right over here.
  • And then when we do a little bit more algebraic manipulation, we're going to see that this thing simplifies to this thing right over here and proves Stokes' theorem for our special case.