Stokes' theorem proof part 3 | Multivariable Calculus | Khan Academy
A1 A2 B1 B2 C1 C2
and
US / ənd /
•UK / ənd, ən,ænd /
- conjunction
- Used to refer to two or more things
- Plus; in addition; on top of that
- Used to introduce an action that follows another
from
US / frəm /
•UK / frʌm, frɑm,frəm /
- preposition
- Starting at a particular place, time or level
out
US / aʊt /
•UK / aʊt /
- 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
US / 'əʊvə(r) /
•UK / ˈovɚ /
- 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
minus
US / ˈmaɪnəs /
•UK / ˈmaɪnəs /
- 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
cross
US / krɒs /
•UK / krɔs, krɑs /
- 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
double
US / 'dʌbl /
•UK / ˈdʌbəl /
- adjecitve
- Concerning or intended for use by two people
- Two times the amount or degree; twice
- Made of two parts that are the same, similar
- noun
- Person that is exactly like someone else
- verb
- To be used for two functions at the same time
- To make or become twice as much
far
US / fɑ:(r) /
•UK / fɑr /
- adjecitve
- A long way away
- Being a long way away
- adverb
- To a large degree or extent
any
US / 'enɪ /
•UK / ˈɛni /
- adjecitve
- One, or some, no matter which
- pronoun
- One (thing) of many; some
come
US / kʌm /
•UK / kʌm /
- verb
- To arrive at a place
- (Of mail) to be delivered
- To move toward someone; go with someone
- To reach a certain point or state
think
US / θɪŋk /
•UK / θɪŋk /
- verb
- To have an idea about something without certainty
- To have an idea, opinion or belief about something
two
US / tu: /
•UK / tu /
- number
- 2
put
US / pʊt /
•UK / pʊt /
- phrasal verb
- to make a light stop shining by pressing or moving a switch
- verb
- To move or place a thing in a particular position
- To rate something in importance or value
- To write or say something in a certain manner
region
US / 'ri:dʒən /
•UK / ˈridʒən /
- noun
- Part of a country, of the world, area, etc.
finally
US / ˈfaɪnəli /
•UK / ˈfaɪnəli /
- adverb
- used especially at the beginning of a sentence to introduce the last point or idea
- After a long time or some difficulty
groundwork
US / ˈgraʊndwɜ:k /
•UK / ˈɡraʊndˌwə:k /
- noun
- Preliminary preparation as a basis or foundation
manipulate
US / mə'nɪpjʊleɪt /
•UK / məˈnɪpjəˌlet /
- verb
- To try to control others to gain an advantage
- To handle or work in a skillful manner
screen
US / skri:n /
•UK / skrin /
- noun
- Wire net in a door or window to keep insects out
- Displays in a theater on which movies are shown
- Flat surface on a computer or TV showing images
- verb
- To hide or block something from being seen
- To examine to find something, e.g. best candidate
few
US / fju: /
•UK / fju /
- pronoun
- Not many, a small number
partial
US / ˈpɑ:ʃl /
•UK / ˈpɑ:rʃl /
- adjecitve
- Giving better treatment to one person than another
- Not complete; being only part of something
for
US / fə(r) /
•UK / fɔr,fə /
- preposition
- Used to show the purpose, or need of something
time
US / taɪm /
•UK / taɪm /
- 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
stuff
US / stʌf /
•UK / stʌf /
- noun
- Generic description for things, materials, objects
- verb
- To push material inside something, with force
exact
US / ɪɡ'zækt /
•UK / ɪɡˈzækt /
- adjecitve
- Completely correct; accurate; specific
- verb
- To get something, sometimes using force
express
US / ɪk'spres /
•UK / ɪkˈsprɛs /
- 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
lower
US / 'ləʊə(r) /
•UK / ˈlaʊɚ, laʊr /
- adjecitve
- In a position more near the bottom
- verb
- To move something downward to the ground
scalar
US / ˈskeɪlə(r) /
•UK / ˈskelɚ, -ˌlɑr /
- adjecitve
- Variable quantity that cannot be resolved into components
well
US / wel /
•UK / wɛl /
- 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
start
US / stɑ:t /
•UK / stɑrt /
- 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
special
US / 'speʃl /
•UK / ˈspɛʃəl /
- adjecitve
- Being additional or extra
- Different, better or greater than normal
- noun
- Heavily discounted sales item
just
US / dʒʌst /
•UK / dʒʌst /
- 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)
how
US / haʊ /
•UK / haʊ /
- 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)
you
US / jʊ /
•UK / ju /
- pronoun
- Person someone is speaking or writing to
- Person or people in general
little
US / 'lɪtl /
•UK / ˈlɪtl /
- 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
US / let /
•UK / lɛt /
- verb
- To allow someone to do something
- Introducing a suggestion to do something together
- To rent a house, etc. to others
the
US / ðə /
•UK / ðə /
- 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
make
US / meɪk /
•UK / mek /
- 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
order
US / 'ɔ:də(r) /
•UK / ˈɔrdɚ /
- 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
across
US / əˈkrɔs /
•UK / əˈkrɔs,əˈkrɑs /
- adverb
- Distance from one side to another
- (Go) from one side to the other of something
this
US / ðɪs /
•UK / ðɪs /
- 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
surface
US / 'sɜ:fɪs /
•UK / ˈsɚfəs /
- 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.
into
US / 'ɪntə /
•UK / ˈɪntu /
- preposition
- Moving or going inside something
- In the same direction as; in the direction of
- So as to become; taking the state of
turn
US / tɜ:n /
•UK / tɚn /
- noun
- Place where changing of direction happens
- Your time or chance to perform a duty
- Your chance to play or do in say a game
- Unexpected change, as in the weather
- verb
- To become (a particular age)
- To become a different quality, color, etc.
- To change the direction of something, e.g. a car
- To move in an opposite direction or position
- To shape metal with a spinning tool
series
US / ˈsɪəri:z /
•UK / ˈsɪriz /
- noun
- Set of stories or articles on a particular subject
- other
- Number of things that happen one after another
already
US / ɔ:lˈredi /
•UK / ɔlˈrɛdi /
- adverb
- Having happened or been done before this time
lay
US / leɪ /
•UK / le /
- adjecitve
- Amateur; not being a trained priest
- verb
- To place a bet on a game outcome
- (Of a bird, etc.) to produce eggs from its body
- To put or set something down in a flat position
- To set out things on (table) to prepare for a meal
will
US / wɪl /
•UK / wɪl /
- 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
plus
US / plʌs /
•UK / plʌs /
- noun
- A benefit or advantage
- preposition
- In addition; add
- In addition to; Indicates a number above
- Having a value greater than zero
side
US / saɪd /
•UK / saɪd /
- 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
component
US / kəmˈpəʊnənt /
•UK / kəmˈponənt /
- adjecitve
- Being a part of something
- noun
- One of the parts that something is made up of
here
US / hɪə(r) /
•UK / hɪr /
- adverb
- In this place; at this location, position or time
try
US / traɪ /
•UK / traɪ /
- 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
actually
US / ˈæktʃuəli /
•UK / ˈæktʃuəli /
- adverb
- Used to add new (often different) information
integral
US / ˈɪntɪgrəl /
•UK / ˈɪntɪɡrəl, ɪnˈtɛɡrəl /
- adjecitve
- Being an essential part of something; necessary
use
US / ju:s /
•UK / ju:s /
- 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
US / raɪt /
•UK / raɪt /
- 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
consider
US / kən'sɪdə(r) /
•UK / kənˈsɪdər /
- verb
- To think carefully about something
true
US / tru: /
•UK / tru: /
- adjecitve
- Agreeing with the facts; not false; real or actual
ago
US / əˈgəʊ /
•UK / əˈgoʊ /
- adverb
- At a specified length of time in the past
figure
US / ˈfiɡə /
•UK / ˈfɪɡjɚ /
- 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
US / weɪ /
•UK / we /
- 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
everything
US / 'evrɪθɪŋ /
•UK / ˈɛvriˌθɪŋ /
- pronoun
- All of the things mentioned
swap
US / swɒp /
•UK / swɑp /
- noun
- A trade or exchange of goods without using money
- verb
- To exchange goods without spending money; trade
they
US / ðeɪ /
•UK / ðe /
- pronoun
- Two or more people, animals, or things
expression
US / ɪk'spreʃn /
•UK / ɪkˈsprɛʃən /
- noun
- Act of making your thoughts and feelings known
- Group of words that have a specific meaning
pretty
US / 'prɪtɪ /
•UK / ˈprɪti /
- adjecitve
- Being attractive to the eye in a simple way
- adverb
- Very; to a great degree
sub
US / sʌb /
•UK / sʌb /
- prefix
- A prefix meaning 'under' or 'instead of'
all
US / ɔ:l /
•UK / ɔl /
- adverb
- Completely; totally
- determiner
- Being every one of something
- pronoun
- 100% of something; the complete amount
take
US / teɪk /
•UK / tek /
- 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
bit
US / bɪt /
•UK / bɪt /
- 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
cancel
US / 'kænsl /
•UK / ˈkænsəl /
- verb
- To end a legal, business, or other agreement
- To stop a planned event from happening
general
US / 'dʒenrəl /
•UK / ˈdʒɛnərəl /
- adjecitve
- Widespread, normal or usual
- noun
- Top ranked officer in the army
rewrite
US / ˌri:'raɪt /
•UK / riˈraɪt /
- verb
- To write again to improve, or correct errors
multiply
US / ˈmʌltɪplaɪ /
•UK / ˈmʌltəˌplaɪ /
- 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
both
US / bəuθ /
•UK / boθ /
- determiner
- Used to refer to two things at the same time
- pronoun
- Used to refer to two things at the same time
show
US / ʃəʊ /
•UK / ʃo /
- noun
- False display of love, emotion or action
- An event for displaying or promoting goods
- Play, opera or theater production
- Presentation that is broadcast on TV or radio
- verb
- To be easily seen or displayed
- To display your emotions or feelings
- To let someone see something; display
- To explain or teach how something is done
- To lead someone somewhere; point them to something
- To prove something to be true, correct
- To reveal or allow something to be seen
with
US / wɪð /
•UK / wɪð, wɪθ /
- 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
clear
US / klɪə(r) /
•UK / klɪr /
- 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
next
US / nekst /
•UK / nɛkst /
- 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
negative
US / 'neɡətɪv /
•UK / ˈnɛɡətɪv /
- adjecitve
- Being harmful, unwanted or unhelpful
- In mathematics, being less than zero
- Focusing on the bad aspects; pessimistic
- noun
- The opposite to a positive electrical charge
- In grammar, containing words such as 'no' or 'not'
- Reply to a question or statement that means 'no'
- Image on camera film used to make a photo
see
US / si: /
•UK / si /
- 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
US / ˈθɪərəm /
•UK / ˈθiərəm, ˈθɪrəm /
- noun
- A statement that can be proved
almost
US / 'ɔ:lməʊst /
•UK / ˈɔlˌmost, ɔlˈmost /
- adverb
- Only a little less than, nearly, not quite
have
US / həv /
•UK / hæv /
- 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
could
US / kəd /
•UK / kʊd /
- other
- Past form of 'can' to mean have ability
- Used to ask for something politely, or to offer
now
US / naʊ /
•UK / naʊ /
- 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
US / 'i:kwəl /
•UK / ˈikwəl /
- adjecitve
- Same in shape, size, or number
- verb
- To add up or be the same as
operate
US / ˈɒpəreɪt /
•UK / ˈɑ:pəreɪt /
- verb
- To control or handle something, such as a machine
- To control or manage something such as a business
- To be working or being used
- To perform surgery on someone
same
US / seɪm /
•UK / sem /
- adjecitve
- Being equal in quality and amount
- Identical; not different
- Not changing; constant
dot
US / dɒt /
•UK / dɑt /
- noun
- Small round mark
- verb
- To place small amounts/things in various places
- To mark something by placing a small, round spot
video
US / 'vɪdɪəʊ /
•UK / ˈvɪdiˌo /
- adjecitve
- Of a moving film, with sound
- noun
- Recording capturing action with sound
- verb
- To make a moving film of, with sound
stoke
US / stəʊk /
•UK / stok /
- verb
- To add fuel or tend a fire to make it stronger
business
US / ˈbɪznəs /
•UK / ˈbɪznɪs /
- noun
- A company formed for making profit
- Matter that has to be dealt with; task; situation
- Activity of working to make a profit
can
US / kæn /
•UK / kən,kæn /
- 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
but
US / bət /
•UK / bʌt,bət /
- conjunction
- Used before you say something different, opposite
vector
US / 'vektə(r) /
•UK / ˈvɛktɚ /
- noun
- Math a quantity that has both size and direction
our
US / 'aʊə(r) /
•UK / aʊr /
- determiner
- Belonging to us
study
US / 'stʌdɪ /
•UK / ˈstʌdi /
- noun
- Room that a person studies or reads in
- Piece of research
- Topic or subject you research or learn about
- verb
- To focus on learning something usually at school
about
US / ə'baʊt /
•UK / əˈbaʊt /
- 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)
care
US / keə(r) /
•UK / kɛr /
- 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
that
US / ðæt /
•UK / ðæt, ðət /
- 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
US / wɒt /
•UK / hwɑt, hwʌt, wɔt, wʌt,hwət, wət /
- 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
first
US / fɜ:st /
•UK / fɚst /
- adjecitve
- 1st; Coming before all others in time or place
- noun
- Something that has not happened before
- Position of the winner of a competition
get
US / ɡet /
•UK / ɡɛt /
- 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
down
US / daʊn /
•UK / daʊn /
- 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
US / rɪ'spekt /
•UK / rɪˈspɛkt /
- 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
US / ðen /
•UK / ðɛn /
- 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
essentially
US / ɪˈsenʃəli /
•UK / ɪˈsenʃəli /
- adverb
- Basically; (said when stating the basic facts)
curl
US / kɜ:l /
•UK / kɚrl /
- 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
least
US / li:st /
•UK / list /
- adverb
- In the smallest or lowest way
- noun
- Something that is of the lowest or smallest degree
right
US / raɪt /
•UK / raɪt /
- 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
value
US / 'vælju: /
•UK / ˈvælju /
- noun
- Importance you attach to something
- Some information expressed as a number
- Stated price of something; worth
- verb
- To place a high importance on something
- To say how much money something is worth