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Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the time cancels out, and only displacement remains.)
One can similarly rewrite the partial derivative () by expressing dV in terms of dS and dT, putting dV equal to zero and solving for the ratio . When one substitutes that expression in the heat capacity ratio expressed as the ratio of the partial derivatives of the entropy above, it follows:
Stated formally, in general, an equation of motion M is a function of the position r of the object, its velocity (the first time derivative of r, v = dr / dt ), and its acceleration (the second derivative of r, a = d 2 r / dt 2 ), and time t. Euclidean vectors in 3D are denoted throughout in bold.
In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance, there is an automorphism of the graph that carries v to x and w to y. Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and ...
Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
where D / Dt is the material derivative operator, u is the flow velocity, ρ is the local fluid density, p is the local pressure, τ is the viscous stress tensor and B represents the sum of the external body forces. The first source term on the right hand side represents vortex stretching.
See motion graphs and derivatives. A large number of fundamental equations in physics involve first or second time derivatives of quantities. Many other fundamental quantities in science are time derivatives of one another: force is the time derivative of momentum; power is the time derivative of energy
The above derivation uses the first and second laws of thermodynamics. The first law of thermodynamics is essentially a definition of heat, i.e. heat is the change in the internal energy of a system that is not caused by a change of the external parameters of the system.