Misplaced Pages

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72-409: As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has

144-496: A ⋅ x ) {\displaystyle \therefore v^{2}=u^{2}+2({\boldsymbol {a}}\cdot {\boldsymbol {x}})} where v = | v | etc. The above equations are valid for both Newtonian mechanics and special relativity . Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on

216-605: A ) ⋅ x = ( 2 a ) ⋅ ( u t + 1 2 a t 2 ) = 2 t ( a ⋅ u ) + a 2 t 2 = v 2 − u 2 {\displaystyle (2{\boldsymbol {a}})\cdot {\boldsymbol {x}}=(2{\boldsymbol {a}})\cdot ({\boldsymbol {u}}t+{\tfrac {1}{2}}{\boldsymbol {a}}t^{2})=2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}=v^{2}-u^{2}} ∴ v 2 = u 2 + 2 (

288-447: A = d v d t . {\displaystyle {\boldsymbol {a}}={\frac {d{\boldsymbol {v}}}{dt}}.} From there, velocity is expressed as the area under an a ( t ) acceleration vs. time graph. As above, this is done using the concept of the integral: v = ∫ a   d t . {\displaystyle {\boldsymbol {v}}=\int {\boldsymbol {a}}\ dt.} In

360-401: A 2-form to a 3-form in R . Define the current two-form as It measures the amount of "stuff" flowing through a surface per unit time in a "stuff fluid" of density ρ = 1 dx ∧ dy ∧ dz moving with local velocity F . Its exterior derivative dj is then given by Velocity Velocity is the speed in combination with the direction of motion of an object . Velocity

432-497: A solenoidal vector field (also known as an incompressible vector field , a divergence-free vector field , or a transverse vector field ) is a vector field v with divergence zero at all points in the field: ∇ ⋅ v = 0. {\displaystyle \nabla \cdot \mathbf {v} =0.} A common way of expressing this property is to say that the field has no sources or sinks . The divergence theorem gives an equivalent integral definition of

504-413: A source-free part B ( r ) . Moreover, these parts are explicitly determined by the respective source densities (see above) and circulation densities (see the article Curl ): For the irrotational part one has with The source-free part, B , can be similarly written: one only has to replace the scalar potential Φ( r ) by a vector potential A ( r ) and the terms −∇Φ by +∇ × A , and

576-417: A Euclidean coordinate system with coordinates x 1 , x 2 , ..., x n , define In the 1D case, F reduces to a regular function, and the divergence reduces to the derivative. For any n , the divergence is a linear operator, and it satisfies the "product rule" for any scalar-valued function φ . One can express the divergence as a particular case of the exterior derivative , which takes

648-417: A constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration. Since the derivative of the position with respect to time gives the change in position (in metres ) divided by the change in time (in seconds ), velocity is measured in metres per second (m/s). Velocity

720-435: A gas at a constant temperature and pressure, the net flux of gas out of any closed surface is zero. The gas may be moving, but the volume rate of gas flowing into any closed surface must equal the volume rate flowing out, so the net flux is zero. Thus the gas velocity has zero divergence everywhere. A field which has zero divergence everywhere is called solenoidal . If the gas is heated only at one point or small region, or

792-405: A negative value. In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source or a sink at a given point. It is a local measure of its "outgoingness" – the extent to which there are more of the field vectors exiting from an infinitesimal region of space than entering it. A point at which the flux is outgoing has positive divergence, and

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864-453: A small tube is introduced which supplies a source of additional gas at one point, the gas there will expand, pushing fluid particles around it outward in all directions. This will cause an outward velocity field throughout the gas, centered on the heated point. Any closed surface enclosing the heated point will have a flux of gas particles passing out of it, so there is positive divergence at that point. However any closed surface not enclosing

936-412: A solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero: where d S {\displaystyle d\mathbf {S} } is the outward normal to each surface element. The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotational and a solenoidal field. The condition of zero divergence

1008-588: A two-dimensional system, where there is an x-axis and a y-axis, corresponding velocity components are defined as v x = d x / d t , {\displaystyle v_{x}=dx/dt,} v y = d y / d t . {\displaystyle v_{y}=dy/dt.} The two-dimensional velocity vector is then defined as v =< v x , v y > {\displaystyle {\textbf {v}}=<v_{x},v_{y}>} . The magnitude of this vector represents speed and

1080-402: A velocity vector, denotes only how fast an object is moving, while velocity indicates both an object's speed and direction. To have a constant velocity , an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at

1152-468: Is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. The drag force, F D {\displaystyle F_{D}} , is dependent on the square of velocity and is given as F D = 1 2 ρ v 2 C D A {\displaystyle F_{D}\,=\,{\tfrac {1}{2}}\,\rho \,v^{2}\,C_{D}\,A} where Escape velocity

1224-429: Is a function of position which depends on the coordinate system. In Cartesian, cylindrical and spherical coordinates, using the same conventions as before, we have ρ = 1 , ρ = r and ρ = r sin θ , respectively. The volume can also be expressed as ρ = | det g a b | {\textstyle \rho ={\sqrt {\left|\det g_{ab}\right|}}} , where g ab

1296-498: Is a fundamental concept in kinematics , the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity : both magnitude and direction are needed to define it. The scalar absolute value ( magnitude ) of velocity is called speed , being a coherent derived unit whose quantity is measured in the SI ( metric system ) as metres per second (m/s or m⋅s ). For example, "5 metres per second"

1368-522: Is a scalar function of x . Since this definition is coordinate-free, it shows that the divergence is the same in any coordinate system . However the above definition is not often used practically to calculate divergence; when the vector field is given in a coordinate system the coordinate definitions below are much simpler to use. A vector field with zero divergence everywhere is called solenoidal – in which case any closed surface has no net flux across it. In three-dimensional Cartesian coordinates,

1440-676: Is a scalar, whereas "5 metres per second east" is a vector. If there is a change in speed, direction or both, then the object is said to be undergoing an acceleration . The average velocity of an object over a period of time is its change in position , Δ s {\displaystyle \Delta s} , divided by the duration of the period, Δ t {\displaystyle \Delta t} , given mathematically as v ¯ = Δ s Δ t . {\displaystyle {\bar {v}}={\frac {\Delta s}{\Delta t}}.} The instantaneous velocity of an object

1512-417: Is always less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, displacement can increase or decrease in magnitude as well as change direction. In terms of a displacement-time ( x vs. t ) graph, the instantaneous velocity (or, simply, velocity) can be thought of as the slope of the tangent line to the curve at any point , and

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1584-410: Is assumed). Expressions of ∇ ⋅ A {\displaystyle \nabla \cdot \mathbf {A} } in cylindrical and spherical coordinates are given in the article del in cylindrical and spherical coordinates . Using Einstein notation we can consider the divergence in general coordinates , which we write as x , …, x , …, x , where n is the number of dimensions of

1656-720: Is defined as v =< v x , v y , v z > {\displaystyle {\textbf {v}}=<v_{x},v_{y},v_{z}>} with its magnitude also representing speed and being determined by | v | = v x 2 + v y 2 + v z 2 . {\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}.} While some textbooks use subscript notation to define Cartesian components of velocity, others use u {\displaystyle u} , v {\displaystyle v} , and w {\displaystyle w} for

1728-501: Is defined as the rate of change of position with respect to time, which may also be referred to as the instantaneous velocity to emphasize the distinction from the average velocity. In some applications the average velocity of an object might be needed, that is to say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, v ( t ) , over some time period Δ t . Average velocity can be calculated as: The average velocity

1800-435: Is defined as the rate of change of position, it is often common to start with an expression for an object's acceleration . As seen by the three green tangent lines in the figure, an object's instantaneous acceleration at a point in time is the slope of the line tangent to the curve of a v ( t ) graph at that point. In other words, instantaneous acceleration is defined as the derivative of velocity with respect to time:

1872-442: Is expanding, so there will be an outward flux of gas through the surface. So the velocity field will have positive divergence everywhere. Similarly, if the gas is cooled, it will contract. There will be more room for gas particles in any volume, so the external pressure of the fluid will cause a net flow of gas volume inward through any closed surface. Therefore, the velocity field has negative divergence everywhere. In contrast, in

1944-483: Is found by the distance formula as | v | = v x 2 + v y 2 . {\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}}}.} In three-dimensional systems where there is an additional z-axis, the corresponding velocity component is defined as v z = d z / d t . {\displaystyle v_{z}=dz/dt.} The three-dimensional velocity vector

2016-635: Is given by the harmonic mean of the speeds v ¯ = n ( 1 v 1 + 1 v 2 + 1 v 3 + ⋯ + 1 v n ) − 1 = n ( ∑ i = 1 n 1 v i ) − 1 . {\displaystyle {\bar {v}}=n\left({1 \over v_{1}}+{1 \over v_{2}}+{1 \over v_{3}}+\dots +{1 \over v_{n}}\right)^{-1}=n\left(\sum _{i=1}^{n}{\frac {1}{v_{i}}}\right)^{-1}.} Although velocity

2088-486: Is known as moment of inertia . If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit , angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion . Solenoidal vector field In vector calculus

2160-407: Is often called a "source" of the field. A point at which the flux is directed inward has negative divergence, and is often called a "sink" of the field. The greater the flux of field through a small surface enclosing a given point, the greater the value of divergence at that point. A point at which there is zero flux through an enclosing surface has zero divergence. The divergence of a vector field

2232-422: Is often illustrated using the simple example of the velocity field of a fluid, a liquid or gas. A moving gas has a velocity , a speed and direction at each point, which can be represented by a vector , so the velocity of the gas forms a vector field . If a gas is heated, it will expand. This will cause a net motion of gas particles outward in all directions. Any closed surface in the gas will enclose gas which

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2304-474: Is position and r ^ {\displaystyle {\hat {\boldsymbol {r}}}} is the radial direction. The transverse speed (or magnitude of the transverse velocity) is the magnitude of the cross product of the unit vector in the radial direction and the velocity vector. It is also the dot product of velocity and transverse direction, or the product of the angular speed ω {\displaystyle \omega } and

2376-658: Is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as: v = ∇ × A {\displaystyle \mathbf {v} =\nabla \times \mathbf {A} } automatically results in the identity (as can be shown, for example, using Cartesian coordinates): ∇ ⋅ v = ∇ ⋅ ( ∇ × A ) = 0. {\displaystyle \nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0.} The converse also holds: for any solenoidal v there exists

2448-467: Is symmetric A ij = A ji then div ⁡ ( A ) = ∇ ⋅ A {\displaystyle \operatorname {div} (\mathbf {A} )=\nabla \cdot \mathbf {A} } . Because of this, often in the literature the two definitions (and symbols div and ∇ ⋅ {\displaystyle \nabla \cdot } ) are used interchangeably (especially in mechanics equations where tensor symmetry

2520-868: Is taken in order to handle the general case where the determinant might be negative, such as in pseudo-Riemannian spaces. The reason for the square-root is a bit subtle: it effectively avoids double-counting as one goes from curved to Cartesian coordinates, and back. The volume (the determinant) can also be understood as the Jacobian of the transformation from Cartesian to curvilinear coordinates, which for n = 3 gives ρ = | ∂ ( x , y , z ) ∂ ( x 1 , x 2 , x 3 ) | {\textstyle \rho =\left|{\frac {\partial (x,y,z)}{\partial (x^{1},x^{2},x^{3})}}\right|} . Some conventions expect all local basis elements to be normalized to unit length, as

2592-409: Is the gravitational constant and g is the gravitational acceleration . The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. This makes "escape velocity" somewhat of a misnomer, as the more correct term would be "escape speed": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of

2664-406: Is the metric tensor . The determinant appears because it provides the appropriate invariant definition of the volume, given a set of vectors. Since the determinant is a scalar quantity which doesn't depend on the indices, these can be suppressed, writing ρ = | det g | {\textstyle \rho ={\sqrt {\left|\det g\right|}}} . The absolute value

2736-769: Is the component of velocity along a circle centered at the origin. v = v T + v R {\displaystyle {\boldsymbol {v}}={\boldsymbol {v}}_{T}+{\boldsymbol {v}}_{R}} where The radial speed (or magnitude of the radial velocity) is the dot product of the velocity vector and the unit vector in the radial direction. v R = v ⋅ r | r | = v ⋅ r ^ {\displaystyle v_{R}={\frac {{\boldsymbol {v}}\cdot {\boldsymbol {r}}}{\left|{\boldsymbol {r}}\right|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {r}}}} where r {\displaystyle {\boldsymbol {r}}}

2808-546: Is the limit average velocity as the time interval approaches zero. At any particular time t , it can be calculated as the derivative of the position with respect to time: v = lim Δ t → 0 Δ s Δ t = d s d t . {\displaystyle {\boldsymbol {v}}=\lim _{{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {s}}}{\Delta t}}={\frac {d{\boldsymbol {s}}}{dt}}.} From this derivative equation, in

2880-418: Is the mass of the object. The kinetic energy of a moving object is dependent on its velocity and is given by the equation E k = 1 2 m v 2 {\displaystyle E_{\text{k}}={\tfrac {1}{2}}mv^{2}} where E k is the kinetic energy. Kinetic energy is a scalar quantity as it depends on the square of the velocity. In fluid dynamics , drag

2952-461: Is the mass times the distance to the origin times the transverse velocity, or equivalently, the mass times the distance squared times the angular speed. The sign convention for angular momentum is the same as that for angular velocity. L = m r v T = m r 2 ω {\displaystyle L=mrv_{T}=mr^{2}\omega } where The expression m r 2 {\displaystyle mr^{2}}

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3024-547: Is the minimum speed a ballistic object needs to escape from a massive body such as Earth. It represents the kinetic energy that, when added to the object's gravitational potential energy (which is always negative), is equal to zero. The general formula for the escape velocity of an object at a distance r from the center of a planet with mass M is v e = 2 G M r = 2 g r , {\displaystyle v_{\text{e}}={\sqrt {\frac {2GM}{r}}}={\sqrt {2gr}},} where G

3096-463: Is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system). The radial and traverse velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The transverse velocity

3168-453: Is the speed of light. Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles. Consider an object A moving with velocity vector v and an object B with velocity vector w ; these absolute velocities are typically expressed in

3240-436: The x {\displaystyle x} -, y {\displaystyle y} -, and z {\displaystyle z} -axes respectively. In polar coordinates , a two-dimensional velocity is described by a radial velocity , defined as the component of velocity away from or toward the origin, and a transverse velocity , perpendicular to the radial one. Both arise from angular velocity , which

3312-430: The curl and reads as follows: or The Laplacian of a scalar field is the divergence of the field's gradient : The divergence of the curl of any vector field (in three dimensions) is equal to zero: If a vector field F with zero divergence is defined on a ball in R , then there exists some vector field G on the ball with F = curl G . For regions in R more topologically complicated than this,

3384-576: The volume element and F are the components of F = F i e i {\displaystyle \mathbf {F} =F^{i}\mathbf {e} _{i}} with respect to the local unnormalized covariant basis (sometimes written as e i = ∂ x / ∂ x i {\displaystyle \mathbf {e} _{i}=\partial \mathbf {x} /\partial x^{i}} ) . The Einstein notation implies summation over i , since it appears as both an upper and lower index. The volume coefficient ρ

3456-1574: The average velocity as the slope of the secant line between two points with t coordinates equal to the boundaries of the time period for the average velocity. If t 1 = t 2 = t 3 = ... = t , then average speed is given by the arithmetic mean of the speeds v ¯ = v 1 + v 2 + v 3 + ⋯ + v n n = 1 n ∑ i = 1 n v i {\displaystyle {\bar {v}}={v_{1}+v_{2}+v_{3}+\dots +v_{n} \over n}={\frac {1}{n}}\sum _{i=1}^{n}{v_{i}}} v ¯ = s 1 + s 2 + s 3 + ⋯ + s n t 1 + t 2 + t 3 + ⋯ + t n = s 1 + s 2 + s 3 + ⋯ + s n s 1 v 1 + s 2 v 2 + s 3 v 3 + ⋯ + s n v n {\displaystyle {\bar {v}}={s_{1}+s_{2}+s_{3}+\dots +s_{n} \over t_{1}+t_{2}+t_{3}+\dots +t_{n}}={{s_{1}+s_{2}+s_{3}+\dots +s_{n}} \over {{s_{1} \over v_{1}}+{s_{2} \over v_{2}}+{s_{3} \over v_{3}}+\dots +{s_{n} \over v_{n}}}}} If s 1 = s 2 = s 3 = ... = s , then average speed

3528-456: The base body as long as it does not intersect with something in its path. In special relativity , the dimensionless Lorentz factor appears frequently, and is given by γ = 1 1 − v 2 c 2 {\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}} where γ is the Lorentz factor and c

3600-402: The concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue to travel at if it stopped accelerating at that moment. While the terms speed and velocity are often colloquially used interchangeably to connote how fast an object is moving, in scientific terms they are different. Speed, the scalar magnitude of

3672-420: The corresponding components of F , and sum the results. Because applying an operator is different from multiplying the components, this is considered an abuse of notation . For a vector expressed in local unit cylindrical coordinates as where e a is the unit vector in direction a , the divergence is The use of local coordinates is vital for the validity of the expression. If we consider x

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3744-411: The divergence of a continuously differentiable vector field F = F x i + F y j + F z k {\displaystyle \mathbf {F} =F_{x}\mathbf {i} +F_{y}\mathbf {j} +F_{z}\mathbf {k} } is defined as the scalar -valued function: Although expressed in terms of coordinates, the result is invariant under rotations , as

3816-457: The domain. Here, the upper index refers to the number of the coordinate or component, so x refers to the second component, and not the quantity x squared. The index variable i is used to refer to an arbitrary component, such as x . The divergence can then be written via the Voss - Weyl formula, as: where ρ {\displaystyle \rho } is the local coefficient of

3888-461: The identity function F ( x ) = x , we find that: In spherical coordinates , with θ the angle with the z axis and φ the rotation around the z axis, and F again written in local unit coordinates, the divergence is Let A be continuously differentiable second-order tensor field defined as follows: the divergence in cartesian coordinate system is a first-order tensor field and can be defined in two ways: and We have If tensor

3960-540: The inertial frame chosen is that in which the latter of the two mentioned objects is in rest. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame. In the one-dimensional case, the velocities are scalars and the equation is either: v rel = v − ( − w ) , {\displaystyle v_{\text{rel}}=v-(-w),} if

4032-523: The last equality with the contravariant element e ^ i {\displaystyle {\hat {\mathbf {e} }}^{i}} , we can conclude that F i = F ^ i / g i i {\textstyle F^{i}={\hat {F}}^{i}/{\sqrt {g_{ii}}}} . After substituting, the formula becomes: See § In curvilinear coordinates for further discussion. The following properties can all be derived from

4104-558: The latter statement might be false (see Poincaré lemma ). The degree of failure of the truth of the statement, measured by the homology of the chain complex serves as a nice quantification of the complicatedness of the underlying region U . These are the beginnings and main motivations of de Rham cohomology . It can be shown that any stationary flux v ( r ) that is twice continuously differentiable in R and vanishes sufficiently fast for | r | → ∞ can be decomposed uniquely into an irrotational part E ( r ) and

4176-477: The one-dimensional case it can be seen that the area under a velocity vs. time ( v vs. t graph) is the displacement, s . In calculus terms, the integral of the velocity function v ( t ) is the displacement function s ( t ) . In the figure, this corresponds to the yellow area under the curve. s = ∫ v   d t . {\displaystyle {\boldsymbol {s}}=\int {\boldsymbol {v}}\ dt.} Although

4248-458: The ordinary differentiation rules of calculus . Most importantly, the divergence is a linear operator , i.e., for all vector fields F and G and all real numbers a and b . There is a product rule of the following type: if φ is a scalar-valued function and F is a vector field, then or in more suggestive notation Another product rule for the cross product of two vector fields F and G in three dimensions involves

4320-426: The physical interpretation suggests. This is because the trace of the Jacobian matrix of an N -dimensional vector field F in N -dimensional space is invariant under any invertible linear transformation. The common notation for the divergence ∇ · F is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product : take the components of the ∇ operator (see del ), apply them to

4392-431: The point will have a constant density of gas inside, so just as many fluid particles are entering as leaving the volume, thus the net flux out of the volume is zero. Therefore, the divergence at any other point is zero. The divergence of a vector field F ( x ) at a point x 0 is defined as the limit of the ratio of the surface integral of F out of the closed surface of a volume V enclosing x 0 to

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4464-787: The position vector and the functions r ( x ) , θ ( x ) , and z ( x ) , which assign the corresponding global cylindrical coordinate to a vector, in general r ( F ( x ) ) ≠ F r ( x ) {\displaystyle r(\mathbf {F} (\mathbf {x} ))\neq F_{r}(\mathbf {x} )} , θ ( F ( x ) ) ≠ F θ ( x ) {\displaystyle \theta (\mathbf {F} (\mathbf {x} ))\neq F_{\theta }(\mathbf {x} )} , and z ( F ( x ) ) ≠ F z ( x ) {\displaystyle z(\mathbf {F} (\mathbf {x} ))\neq F_{z}(\mathbf {x} )} . In particular, if we consider

4536-754: The radius (the magnitude of the position). v T = | r × v | | r | = v ⋅ t ^ = ω | r | {\displaystyle v_{T}={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {t}}}=\omega |{\boldsymbol {r}}|} such that ω = | r × v | | r | 2 . {\displaystyle \omega ={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|^{2}}}.} Angular momentum in scalar form

4608-710: The same inertial reference frame . Then, the velocity of object A relative to object B is defined as the difference of the two velocity vectors: v A  relative to  B = v − w {\displaystyle {\boldsymbol {v}}_{A{\text{ relative to }}B}={\boldsymbol {v}}-{\boldsymbol {w}}} Similarly, the relative velocity of object B moving with velocity w , relative to object A moving with velocity v is: v B  relative to  A = w − v {\displaystyle {\boldsymbol {v}}_{B{\text{ relative to }}A}={\boldsymbol {w}}-{\boldsymbol {v}}} Usually,

4680-421: The source density div v by the circulation density ∇ × v . This "decomposition theorem" is a by-product of the stationary case of electrodynamics . It is a special case of the more general Helmholtz decomposition , which works in dimensions greater than three as well. The divergence of a vector field can be defined in any finite number n {\displaystyle n} of dimensions. If in

4752-432: The special case of constant acceleration, velocity can be studied using the suvat equations . By considering a as being equal to some arbitrary constant vector, this shows v = u + a t {\displaystyle {\boldsymbol {v}}={\boldsymbol {u}}+{\boldsymbol {a}}t} with v as the velocity at time t and u as the velocity at time t = 0 . By combining this equation with

4824-415: The suvat equation x = u t + a t /2 , it is possible to relate the displacement and the average velocity by x = ( u + v ) 2 t = v ¯ t . {\displaystyle {\boldsymbol {x}}={\frac {({\boldsymbol {u}}+{\boldsymbol {v}})}{2}}t={\boldsymbol {\bar {v}}}t.} It is also possible to derive an expression for

4896-401: The two objects are moving in opposite directions, or: v rel = v − ( + w ) , {\displaystyle v_{\text{rel}}=v-(+w),} if the two objects are moving in the same direction. In multi-dimensional Cartesian coordinate systems , velocity is broken up into components that correspond with each dimensional axis of the coordinate system. In

4968-555: The value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words, only relative velocity can be calculated. In classical mechanics, Newton's second law defines momentum , p, as a vector that is the product of an object's mass and velocity, given mathematically as p = m v {\displaystyle {\boldsymbol {p}}=m{\boldsymbol {v}}} where m

5040-654: The velocity independent of time, known as the Torricelli equation , as follows: v 2 = v ⋅ v = ( u + a t ) ⋅ ( u + a t ) = u 2 + 2 t ( a ⋅ u ) + a 2 t 2 {\displaystyle v^{2}={\boldsymbol {v}}\cdot {\boldsymbol {v}}=({\boldsymbol {u}}+{\boldsymbol {a}}t)\cdot ({\boldsymbol {u}}+{\boldsymbol {a}}t)=u^{2}+2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}} ( 2

5112-449: The volume of V , as V shrinks to zero where | V | is the volume of V , S ( V ) is the boundary of V , and n ^ {\displaystyle \mathbf {\hat {n}} } is the outward unit normal to that surface. It can be shown that the above limit always converges to the same value for any sequence of volumes that contain x 0 and approach zero volume. The result, div F ,

5184-424: Was done in the previous sections. If we write e ^ i {\displaystyle {\hat {\mathbf {e} }}_{i}} for the normalized basis, and F ^ i {\displaystyle {\hat {F}}^{i}} for the components of F with respect to it, we have that using one of the properties of the metric tensor. By dotting both sides of

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