Given
[tex]\vec F = (4xz+y^2)\,\vec\imath + 2xy\,\vec\jmath + 2x^2\,\vec k[/tex]
its curl would be
[tex]\nabla\times\vec F = \left(\dfrac{\partial(2xy)}{\partial z} - \dfrac{\partial(2x^2)}{\partial y}\right)\,\vec\imath - \left(\dfrac{\partial(4xz+y^2)}{\partial z} - \dfrac{\partial(2x^2)}{\partial x}\right)\,\vec\jmath + \left(\dfrac{\partial(4xz+y^2)}{\partial y} - \dfrac{\partial(2xy)}{\partial x}\right)\,\vec k[/tex]
which reduces to the zero vector. Since the curl is zero, and [tex]\vec F[/tex] doesn't have any singularities, [tex]\vec F[/tex] is indeed a gradient field.
To determine what it is a gradient of, we look for a scalar function f(x, y, z) such that [tex]\nabla f = \vec F[/tex]. This entails solving for f such that
[tex]\dfrac{\partial f}{\partial x} = 4xz+y^2 \\\\ \dfrac{\partial f}{\partial y} = 2xy \\\\ \dfrac{\partial f}{\partial z} = 2x^2[/tex]
Integrate both sides of the first equation with respect to x, which gives
[tex]f(x,y,z) = 2x^2z+xy^2 + g(y,z)[/tex]
Differentiate both sides of this with respect to y :
[tex]\dfrac{\partial f}{\partial y} = 2xy = 2xy+\dfrac{\partial g}{\partial y} \\\\ \implies \dfrac{\partial g}{\partial y} = 0 \implies g(y,z) = h(z)[/tex]
Differentiate f with respect to z :
[tex]\dfrac{\partial f}{\partial z} = 2x^2 = 2x^2 + \dfrac{\mathrm dh}{\mathrm dz} \\\\ \implies \dfrac{\mathrm dh}{\mathrm dz} = 0\implies h(z) = C[/tex]
So it turns out that [tex]\vec F[/tex] is the gradient of
[tex]f(x,y,z) = 2x^2z+xy^2+C[/tex]
Name another way that energy is made
Answer:
The three major categories of energy for electricity generation are fossil fuels (coal, natural gas, and petroleum), nuclear energy, and renewable energy sources. Most electricity is generated with steam turbines using fossil fuels, nuclear, biomass, geothermal, and solar thermal energy.
Explanation:
Solar
Wind
Geothermal
Hydrogen
TidaWave
Hydroelectric
Biomass
what is matter made of ?
Explanation:
Matter is anything occupies space and has mass
Matter is made up small tiny particles called atom
Atom contains:-
Proton
Neutron
Electron
A lad, waiting for his friend walks in the sidewalk, in front of her house, from the front door, first, he moves towards the Positive x-axis, 5 m, then goes back, 6 m. What is his total displacement from his original position? Displacement is a vector quantity whilst, distance is a scalar quantity.
His total displacement from his original position is -1 m
We know that total displacement of an object from a position x to a position x', d = final position - initial position.
d = x' - x
If we assume the lad's initial position in front of her house is x = 0 m. The lad then moves towards the positive x-axis, 5 m. He then ends up at x' = 5 m. He then finally goes back 6 m.
Since displacement = final position - initial position, and his displacement is d' = -6 m (since he moves in the negative x - direction or moves back) from his initial position of x' = 5 m.
His final position, x" after moving back 6 m is gotten from
x" - x' = -6 m
x" = -6 + x'
x" = -6 + 5
x" = -1 m
Thus, his total displacement from his original position is
d = final position - initial position
d = x" - x
d = -1 m - 0 m
d = -1 m
So, his total displacement from his original position is -1 m
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Based on what location on the periodic table which element is metal
Answer:
The metals are to the left of the line (except for hydrogen, which is a nonmetal), the nonmetals are to the right of the line, and the elements immediately adjacent to the line are the metalloids.
Question 9 of 10
Which of the following is equal to the area under a velocity-time graph?
A. Velocity
B. Displacement
C. Force
D. Accelertion
SUBMIT
Displacement is equal to the area under a velocity-time graph.
A car is traveling at a constant speed on the highway. Its tires have a diameter of 65.0 cm and are rolling without sliding or slipping. If the angular speed of the tires is 49.0 rad/s , what is the speed of the car, in SI units
Answer:
Explanation:
The car is rolling without slipping so Vcm= R×ω = 0.325×49 = 16
A solid sphere and a solid cylinder , both uniform and having the same mass and radius, are released from rest at the same height on a ramp. If they roll down the ramp without slipping, which one will have the lower forward velocity at the bottom of the ramp?
a. The solid sphere
b. The solid cylinder
c. Both will have the same forward velocity
Answer:
I (cylinder) = 1/2 M R^2
I (sphere) = 2/5 M R^2
The equation of motion shows
I * alpha = M g R sin theta = const
An increase in the moment of inertia will result in a smaller acceleration
The cylinder will have the smallest forward velocity because it will arrive at the bottom after the sphere
How to use calculus in physics ?
Explanation:
In Physics, many quanitties are the function of other quanitities.
For instance in kinematics, where the displacement (s), position (x), velocity (v) and time (t) of a particle are related.
Suppose the position of a particle can be descirbed as a function of time, i.e.
[tex]x=f(t)[/tex]
then since velocity is the rate of change of position, we write
[tex]v=\dfrac{d}{dt}f(t)=f'(t)[/tex]
which is the derivative of the poisiton/ time function.
On the other hand, the displacement of the particle is the sum of position changes over time, so the displacement can be defined by
[tex]s=\int^{t_1}_{t_0}f(t)dt[/tex]
which is the integration of postion/ time function.
An engineering company needs to solve a design problem.
It is designing a high-speed train to transport passengers
between Boston and Atlanta.
Which statement is a criterion for this project?
O A. The train must not take more than six years to build.
B. The train must not have wheels with bearings that need
lubrication,
O C. The train must cost less than $150 million to design.
D. The train must be able to maintain a speed of 250 km/h.
The engineering company must consider the speed of the train in the design.
The correct answer is option D.
To design a high speed train to transport passengers between Boston and Atlanta, the engineering company must include the following design criterion in the project:
the speed of the train: this must be high in order to satisfy the high-speed criterion in the statement of the problem.Thus, the only statement in all the given options that considered speed of the train is option D.
"D. The train must be able to maintain a speed of 250 km/h."
Learn more here: https://brainly.com/question/16173068
Answer:
The train must provide a smooth ride for passengers between bosten and atlanta
Explanation:
Diamond has a density of 3.26 g/cm3
. What
is the mass of a diamond that has a volume of
0.214 cm3
?
Answer in units of g
Explanation:
Hey there!
According to the question;
Density of diamond (d) = 3.26g/cm³
And Volume (v) = 0.214 cm³
Mass(m) = ?
From the definition of density;
d = m/v
Or, 3.26 g/cm³ = m/(0.214 cm³)
or, 3.26 g/cm³ * 0.214 cm³ = m
Therefore, mass of the diamond is 0.69764 g.
Hope it helps!
LESSON-THE PLANT Fairy
1) Fill in the blanks using given options.
( Grass, tree, Mehndi, shapes and colour)
a) Plants with thick or thin trunk are called as
b) Leaves of different plants have different
c) Didi was sitting on the
d) is a bush.
wite a paragraph explaining the difference between things that have matter and things that don't have matter.
Answer: Matter is how much space or opacity an object takes up. In short anything that take up space. Things like balls, trees, and even people are all made of matter. Mass is how much matter a object has. Air also has mass also, even though we can't see it. Things like cars, buildings, even planets have mass.
Explanation: Paragraphs are sometimes 4-6 sentences.
prove the conservation of linear momentum
Answer:
force experienced by A and B are equal and opposite, that is total momentum before collision is equal to total momentum after collision if no external forces act on them which proves the principle of conservation of linear momentum.
Which type of bond and how many bonds would occur as carbon dioxide (CO2) is formed from carbon in Group IVA and oxygen in Group VIA?
A. Three metallic bonds
B. two ionic bonds
C. One covalent bond
D. Four covalent bonds
Answer:
The answer is D. Four covalent bounds
Will we ever be able to know if there are other universes like our own? what do you think,
some say it is impossible to see beyond our own universe. but hey, we used to think that the earth was flat and look at us now, planing to go to mars.
however, it is impossible to know what kind of advancements lie ahead for science.
Answer:
I think it's possible to know if more universes exist like how we found out that more galaxies exist.
Prove that s=ut+½at²
Explanation:
Distance travelled = Area under the line
= ut + ½ (v-u)t
Acceleration (a) = (v-u)/t and so (v-u) = at
Therefore,
Distance travelled (s) = ut + ½ (v-u)t = ut + ½ (at)t = ut + ½ at²
Thus,proved.
A car speeds up from 0.0 m/s to 14.0 m/s
in 3.25 s.
The acceleration of the car is:
A. 14.0 m/s2
B. 4.31 m/s2
C. 45.5 m/s2
D. 10.8 m/s2
Explanation:
Its speed is zero at the beginning of the 3.25 s and 14 m/s at the end.
I'm thinking you probably wanted to find the car's acceleration.
Acceleration =(change in speed) / (time for the change).
Acceleration =(14m/s )/ (3.25s) .
Acceleration = 4.31m/s² .
I hope it will help you so.
Have a great day.
Be safe and be happy.
Differences between law of flotation and Archimedes's principle
Answer:
Differences between law of flotation and Archimedes's principle
here
Explanation:
2 1.1.4 Quiz: Introduction to Physical Science
Question 2 of 10
Which expression correctly describes energy using SI units?
O A. 1 J = 1 kg-m/s2
O B. 1 J = 1 kg.m/s
C. 1 J = 1 kg:m2/s2
D. 1 J = 1 kg.m/s
SUBMIT
Answer:
A. 1 J= 1 kg•m²/s² :)
Explanation:
Vector A has a magnitude of 75.0 cm and points at 30° above the positive x-axis. Vector B has a magnitude of 25.0 cm and points along the negative x-axis. Vector C⃗ has a magnitude of 40.0 cm and points at 45° below the negative x-axis.
(a) Determine the x and y components of Vector A
(b) Determine the x and y components of Vector B .
(c) Determine the x and y components of Vector C .
(d) Determine x and y components of the sum of these three vectors.
(e) Determine the magnitude of the sum of these three vectors.
(f) Determine the direction of the sum of these three vectors.
(a)
A = (75.0 cm) (cos(30°) i + sin(30°) j)
A = (75.0 cm) (√3/2 i + 1/2 j)
A ≈ (64.95 i + 37.5 j) cm
The x component is the coefficient of the i unit vector, while the y component is the coefficient of the j unit vector.
(b) "points along the negative axis" in other words means that B makes an angle of 180° with the positive x-axis in the counterclockwise direction. In particular this tells you that B is parallel to the x-axis, so its y component would be zero.
B = (25.0 cm) (cos(180°) i + sin(180°) j)
B = (25.0 cm) (-1 i + 0 j)
B = (-25.0 i) cm
(c) If the negative x-axis corresponds to 180°, then 45° below this would make an angle of 180° + 45° = 225° with the positive x-axis.
C = (40.0 cm) (cos(225°) i + sin(225°) j)
C = (40.0 cm) (-1/√2 i - 1/√2 j)
C ≈ (-28.28 i - 28.28 j) cm
(d) The sum A + B + C has components that are the sums of the components of each of A, B, and C.
A + B + C ≈ (11.67 i + 9.22 j) cm
(e) The magnitude of the vector sum is
||A + B + C|| = √((11.67 cm)² + (9.22 cm)²) ≈ 14.87 cm
(f) Since both components of the vector sum are positive, you know that it's a vector that terminates in the first quadrant of the x,y-plane, so it makes an angle with the positive x-axis of
arctan(9.22/11.67) ≈ 38.3°
what is the function of thrombocytes
Answer:
the function of the thrombocytes is to clott blood
In which version does George utilize some of the steps of Effective Conflict Management?
Answer:
The Effective Cooperative Conflict management consists of a series of steps to take when two or more people have a clash or opposing interests that in some way get in the way of the action of the other. These steps consist of the following:
1) manage emotions,
2) manage feedback
3) manage a goal.
Convert 17800 seconds into day
Answer:
the answer is 0.206 day
hope this helps!
Answer:
17800 Second (s) = 0.206019 Day (d)
If the mass of an object is 5 kg and the velocity is 8 m/s, what is the momentum?
Answer:
40 kg•m/s
Explanation:
momentum (p) = mass (kg) × velocity (m/s)
How far will a bus travel if it averages a speed of 65 km/h for 7 hours?
What are you solving for?
What’s the equation?
Show work!
distance= speed*time
distance=65*7=455km
_________statistics are measures that summarize the characteristics of a sample
Answer:
Explanation:
Descriptive statistics statistics are measures that summarize the characteristics of a sample.
Hope it helps
vận tốc là 12km|h tgian 2h hỏi quãng đường
Convert 4 centimeters to meters using dimensional analysis and scientific notation. Please go step by step as I am having difficulty with this subject.
Answer:
[tex]thank \: you[/tex]
A rock is thrown straight up from the edge of a cliff. The rock reaches the maximum height of 15 m above the edge and then falls down to the bottom of the cliff 35 m below the cliff. What is the traveled distance of the rock?
A. 30 m B. 35 m C. 50 m D. 65 m
Answer:
The distance travelled by the rock is 65m
The rock is thrown 15 m up from the edge of the cliff and it have to travel 15 m down to reach the cliff and go to bottom of the cliff by 35 m. Thus, the total distance travelled by the rock is 65 m.
What is distance?Distance is a physical quantity used to measure how far an object is travelling. It is a scalar quantity and is characterised by magnitude only. Distance travelled is sometimes substituted with displacement. Both have the units in cm, m, feet, kilometer etc.
Distance in physics is the product of the speed of a moving body and the time taken complete the travel.
Distance = speed × time.
It is given that, the rock is thrown straight up from the edge of a cliff and it reaches upto 15 m. Then it have to return back this 15 m down to reach the edge. From there it goes 35 m below.
Thus, total distance = 15 + 15 + 35 = 65 m.
Therefore, the total distance travelled by the rock is 65 m.
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A probability of breaking a tree is more than a short one during storm
Answer:
due to inertia of motion