Differential Equations

To find the longest path in a graph, you can use dynamic programming or backtracking techniques, particularly in Directed Acyclic Graphs (DAGs). First, perform a topological sort of the graph. Then, initialize a distance array and relax the edges in topological order, updating the longest distance to each vertex. For undirected graphs or those with cycles, the problem is NP-hard, and heuristic or approximation algorithms may be necessary.

60x680

1cm

Exact differentials refer to changes in a function that can be expressed as the total differential of that function, meaning there exists a scalar function whose differential equals the given expression. Inexact differentials, on the other hand, cannot be derived from a single function and typically arise in contexts like thermodynamics, where they represent changes in quantities that are path-dependent. In mathematical terms, an exact differential ( df ) satisfies the condition ( \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y} ), while inexact differentials, such as ( dQ ) or ( dW ), do not meet this criterion.

It is the solution of a differential equation without there being any restrictions on the variables (No boundary conditions are given). Presence of arbitrary constants indicates a general solution, the number of arbitrary constants depending on the order of the differential equation.

Try the link. It will give you the basics.

.5333 repeated

24 July

109

2.0833

The Cauchy-Riemann equations are fundamental in complex analysis, providing conditions for a function to be holomorphic, meaning it is complex differentiable. These equations are essential in various fields, including fluid dynamics, where they describe potential flow, and in electrical engineering for analyzing electromagnetic fields. Additionally, they are used in conformal mapping, which allows for the transformation of complex shapes in a way that preserves angles, facilitating the solution of physical problems in engineering and physics.

"Charge?"

The field produced by an inductor exists ONLY while the current flow is changing,

its pressure between suction and discharge flow...

(cos(pi x) + sin(pi y) )^8 = 44

differentiate both sides with respect to x

8 ( cos(pi x) + sin (pi y ) )^7 d/dx ( cos(pi x) + sin (pi y) = 0

8 ( cos(pi x) + sin (pi y ) )^7 (-sin (pi x) pi + cos (pi y) pi dy/dx ) = 0

8 ( cos(pi x) + sin (pi y ) )^7 (pi cos(pi y) dy/dx - pi sin (pi x) ) = 0

cos(pi y) dy/dx - pi sin(pi x) = 0

cos(pi y) dy/dx = sin(pi x)

dy/dx = sin (pi x) / cos(pi y)

Potential differential, often referred to as the electrochemical potential difference, is the difference in electric potential between two points, which drives the movement of charged particles, such as ions, across a membrane. It is crucial in biological systems, particularly in neurons and muscle cells, as it governs processes like action potentials and synaptic transmission. Understanding potential differentials is essential for studying cellular communication, metabolism, and overall physiological functions. Additionally, it has applications in fields like bioengineering and pharmacology, influencing drug delivery and the development of medical devices.

It means that the dependent variable and all its derivatives are multiplied by constants only, not by themselves nor by functions containing the independent variable..

For example, (dy/dx) + xy = 0 is non-linear

but (dy/dx) + y = (x^2)coswx is linear.

(Note that it doesnt matter how the function of the independent variable is)

unsquaring a number is to chage it back

Heat loss due to change in temperature:

Q = mc(T2-T1)

Heat loss due to change in phase:

Q = mL

c and L are constants that are specific to each compound at certain temperatures. For water, we usually take c to be 4186 J/(kg*K).

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A gigaelectonvolt is 1 billion electronvolts.