{"version":"https://jsonfeed.org/version/1","title":"Quantum Equations","home_page_url":"https://quantum-equations.com","description":"Quantum Equations","author":{"name":"Quantum Equations"},"items":[{"id":"https://quantum-equations.com/equation-list/f/flow-of-water-uniform-horizontal-cylinder","html_content":"

NOTE: Use only for a circular container that is extending horizontally.

","url":"https://quantum-equations.com/equation-list/f/flow-of-water-uniform-horizontal-cylinder","title":"Flow of Water (Uniform Horizontal Cylinder)","summary":"NOTE: Use only for a circular container that is extending horizontally.","date_modified":"2026-05-28T02:26:28Z"},{"id":"https://quantum-equations.com/equation-list/f/flow-of-water-through-an-opening","html_content":"

NOTE: Use only for a circular opening that the water is flowing through.

","url":"https://quantum-equations.com/equation-list/f/flow-of-water-through-an-opening","title":"Flow of Water Through an Opening ","summary":"NOTE: Use only for a circular opening that the water is flowing through.","date_modified":"2026-05-25T00:36:00Z"},{"id":"https://quantum-equations.com/equation-list/f/dissolving-material","html_content":"

(Note: The constant k has been integrated into the equation)

","url":"https://quantum-equations.com/equation-list/f/dissolving-material","title":"Dissolving Material","summary":"(Note: The constant k has been integrated into the equation)","date_modified":"2026-05-23T20:17:35Z"},{"id":"https://quantum-equations.com/equation-list/f/combining-elements","html_content":"

(Note: [f(x) = green graph] for this example, when visible with the graph [y sub 1(x) = orange graph] the data to use is f(x)). The equation for y sub 1(x) is used when even reaction distribution is encountered and when...

","url":"https://quantum-equations.com/equation-list/f/combining-elements","title":"Combining Elements ","summary":"(Note: [f(x) = green graph] for this example, when visible with the graph [y sub 1(x) = orange graph] the data to use is f(x)). The equation for y sub 1(x) is used when even reaction distribution is encountered and when...","date_modified":"2026-05-23T19:54:57Z"},{"id":"https://quantum-equations.com/equation-list/f/growth","html_content":"

Equation f is amount over time.

","url":"https://quantum-equations.com/equation-list/f/growth","title":"Growth","summary":"Equation f is amount over time.","date_modified":"2026-04-24T22:03:12Z"},{"id":"https://quantum-equations.com/equation-list/f/decay","html_content":"

Equation f is amount remaining in percent after time has passed.

","url":"https://quantum-equations.com/equation-list/f/decay","title":"Decay","summary":"Equation f is amount remaining in percent after time has passed.","date_modified":"2026-04-24T21:59:58Z"},{"id":"https://quantum-equations.com/equation-list/f/submerged-object-heat-transfer","html_content":"

NOTE: This Equation DOES NOT Account for Volatile Rapid Expansion of the Medium if the BodyTemp is Very Large.

","url":"https://quantum-equations.com/equation-list/f/submerged-object-heat-transfer","title":"Submerged Object Heat Transfer","summary":"NOTE: This Equation DOES NOT Account for Volatile Rapid Expansion of the Medium if the BodyTemp is Very Large.","date_modified":"2026-04-21T21:26:45Z"},{"id":"https://quantum-equations.com/equation-list/f/temperature-at-time","html_content":"

NOTE: This Equation DOES NOT Account for Volatile Rapid Expansion of the Medium if the BodyTemp is Very Large.

","url":"https://quantum-equations.com/equation-list/f/temperature-at-time","title":"Temperature At Time","summary":"NOTE: This Equation DOES NOT Account for Volatile Rapid Expansion of the Medium if the BodyTemp is Very Large.","date_modified":"2026-04-17T18:53:30Z"},{"id":"https://quantum-equations.com/equation-list/f/temperature-over-time","html_content":"

NOTE: This Equation DOES NOT Account for Volatile Rapid Expansion of the Medium if the BodyTemp is Very Large.

","url":"https://quantum-equations.com/equation-list/f/temperature-over-time","title":"Temperature Over Time","summary":"NOTE: This Equation DOES NOT Account for Volatile Rapid Expansion of the Medium if the BodyTemp is Very Large.","date_modified":"2026-04-17T17:57:09Z"},{"id":"https://quantum-equations.com/equation-list/f/solution-concentration-rate-in-%E2%89%A0-rate-out","html_content":"

Note: This equation is only for rate in ≠ rate out, it will not calculate correctly if used otherwise.

","url":"https://quantum-equations.com/equation-list/f/solution-concentration-rate-in-%E2%89%A0-rate-out","title":"Solution Concentration rate in ≠ rate out","summary":"Note: This equation is only for rate in ≠ rate out, it will not calculate correctly if used otherwise.","date_modified":"2026-04-15T03:13:22Z"},{"id":"https://quantum-equations.com/equation-list/f/solution-concentration-rate-in-rate-out","html_content":"

Note: This equation is only for rate in = rate out, it will not calculate correctly if using different rates.

","url":"https://quantum-equations.com/equation-list/f/solution-concentration-rate-in-rate-out","title":"Solution Concentration Rate in = Rate out","summary":"Note: This equation is only for rate in = rate out, it will not calculate correctly if using different rates.","date_modified":"2026-04-15T03:02:46Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-r-sin%CE%B8-k","html_content":"

Equation: -ln| secΘ +tanΘ| + c

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-r-sin%CE%B8-k","title":"Orthogonal to r = sinΘ + k","summary":"Equation: -ln| secΘ +tanΘ| + c","date_modified":"2026-04-13T03:48:00Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-r-e%5Ek-%CE%B8","html_content":"

Equation: e^(c - Θ²)^(1/2)

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-r-e%5Ek-%CE%B8","title":"Orthogonal to r = e^(k * Θ)","summary":"Equation: e^(c - Θ²)^(1/2)","date_modified":"2026-04-13T03:16:21Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-r%5En-sinn-%CE%B8-k","html_content":"

Equation: r = (c * sec(n * Θ))^1/n

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-r%5En-sinn-%CE%B8-k","title":"Orthogonal to r^n * sin(n * Θ) = k","summary":"Equation: r = (c * sec(n * Θ))^1/n","date_modified":"2026-04-13T02:47:13Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-r%C2%B2-k-cos2%CE%B8","html_content":"

Equation: c * (sin(2 * Θ))^1/2

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-r%C2%B2-k-cos2%CE%B8","title":"Orthogonal to r² = k * cos(2Θ)","summary":"Equation: c * (sin(2 * Θ))^1/2","date_modified":"2026-04-12T23:42:48Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-x%C2%B2---y%C2%B2-k%C2%B2","html_content":"

Equation: x* y = c

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-x%C2%B2---y%C2%B2-k%C2%B2","title":"Orthogonal to x² - y² = k²","summary":"Equation: x* y = c","date_modified":"2026-04-12T22:04:53Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-y-k-x%C2%B2","html_content":"

Equation: y = ±(c + (x² / 2))^½

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-y-k-x%C2%B2","title":"Orthogonal to y = k / x²","summary":"Equation: y = ±(c + (x² / 2))^½","date_modified":"2026-04-10T19:18:55Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-y-k-x--1-x","html_content":"

Equation: (-x³ / 3) + c

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-y-k-x--1-x","title":"Orthogonal to y = ( k* x -1) / x","summary":"Equation: (-x³ / 3) + c","date_modified":"2026-04-10T17:52:36Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-y%C2%B2-4kx","html_content":"

Equation: (y² / 2) + x² = c

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-y%C2%B2-4kx","title":"Orthogonal to y² = 4kx","summary":"Equation: (y² / 2) + x² = c","date_modified":"2026-04-10T17:35:52Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-siny-k-e%5Ex%C2%B2","html_content":"

Equation: x = c * cos²y

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-siny-k-e%5Ex%C2%B2","title":"Orthogonal to sin(y) = k * e^x²","summary":"Equation: x = c * cos²y","date_modified":"2026-04-09T16:05:36Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-e%5Ex-cosy-k","html_content":"

Equation: e^x * sin(y) = c

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-e%5Ex-cosy-k","title":"Orthogonal to e^x * cos(y) = k","summary":"Equation: e^x * sin(y) = c","date_modified":"2026-04-09T13:50:10Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-x---y-k-e%5Ex","html_content":"

Equation: c * e^y = 2 + y - x

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-x---y-k-e%5Ex","title":"Orthogonal to (x - y) = k* e^x","summary":"Equation: c * e^y = 2 + y - x","date_modified":"2026-04-09T13:01:45Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-x---k%C2%B2-y%C2%B2-k%C2%B2","html_content":"

Equation: y² + x² = y * c

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-x---k%C2%B2-y%C2%B2-k%C2%B2","title":"Orthogonal to (x - k)² + y² = k²","summary":"Equation: y² + x² = y * c","date_modified":"2026-04-09T12:45:18Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-x%C2%B2-y---k%C2%B2-k%C2%B2","html_content":"

Equation: y² + x² = x * c

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-x%C2%B2-y---k%C2%B2-k%C2%B2","title":"Orthogonal to x² + (y - k)² = k²","summary":"Equation: y² + x² = x * c","date_modified":"2026-04-09T12:07:46Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-x%5E2-2xy---y%5E2-k","html_content":"

Equation: y^2 + (2 * x * y) - x^2 = c

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-x%5E2-2xy---y%5E2-k","title":"Orthogonal to x^2 + 2xy - y^2 = k","summary":"Equation: y^2 + (2 * x * y) - x^2 = c","date_modified":"2026-04-09T11:36:14Z"},{"id":"https://quantum-equations.com/equation-list/f/isogonal-to-y%5E2-2xy---x%5E2","html_content":"

Equation: x * y = c

","url":"https://quantum-equations.com/equation-list/f/isogonal-to-y%5E2-2xy---x%5E2","title":"Isogonal to y^2 + 2xy - x^2","summary":"Equation: x * y = c","date_modified":"2026-04-08T02:43:53Z"},{"id":"https://quantum-equations.com/equation-list/f/orthogonal-to-y-cx%5E5","html_content":"

Equation: (5 * y^2) + x^2 = k

","url":"https://quantum-equations.com/equation-list/f/orthogonal-to-y-cx%5E5","title":"Orthogonal to y = cx^5","summary":"Equation: (5 * y^2) + x^2 = k","date_modified":"2026-04-08T02:07:18Z"},{"id":"https://quantum-equations.com/equation-list/f/isogonal-to-y-ax%5E2","html_content":"

Equation: ln|(2* y^2) - (x * y) + x^2| + (6 / (7)^(1/2)) * arctan((4*y - x) / (((7)^(1/2)) * x)) = c

","url":"https://quantum-equations.com/equation-list/f/isogonal-to-y-ax%5E2","title":"Isogonal to y = ax^2 ","summary":"Equation: ln|(2* y^2) - (x * y) + x^2| + (6 / (7)^(1/2)) * arctan((4*y - x) / (((7)^(1/2)) * x)) = c","date_modified":"2026-04-08T01:42:35Z"},{"id":"https://quantum-equations.com/equation-list/f/area-between-arc-and-radius-vector-is-half-arc-length","html_content":"

Equation: r = sec(Θ + c)

","url":"https://quantum-equations.com/equation-list/f/area-between-arc-and-radius-vector-is-half-arc-length","title":"Area Between Arc and Radius Vector is Half Arc Length","summary":"Equation: r = sec(Θ + c)","date_modified":"2026-04-07T15:28:36Z"},{"id":"https://quantum-equations.com/equation-list/f/radius-vector-and-tangent-intersect-equals-k-times-polar-angle","html_content":"

Equation: r = [c * sin(k * Θ)]^(1 / k)

","url":"https://quantum-equations.com/equation-list/f/radius-vector-and-tangent-intersect-equals-k-times-polar-angle","title":"Radius Vector and Tangent Intersect Equals k Times Polar Angle","summary":"Equation: r = [c * sin(k * Θ)]^(1 / k)","date_modified":"2026-04-07T06:42:48Z"},{"id":"https://quantum-equations.com/equation-list/f/radius-vector-and-tangent-intersection-form-constant-angle","html_content":"

Equation: r = c * e^(Θ* cotß)

","url":"https://quantum-equations.com/equation-list/f/radius-vector-and-tangent-intersection-form-constant-angle","title":"Radius Vector and Tangent Intersection Form Constant Angle","summary":"Equation: r = c * e^(Θ* cotß)","date_modified":"2026-04-05T02:00:12Z"},{"id":"https://quantum-equations.com/equation-list/f/the-integral-f-rotated-about-either-axis-is-equal","html_content":"

Note: f(x) rotated about the x-axis between x = a_x and x = b_x, and f(x) rotated about the y-axis between y = a_y and b_y. Where a and b are the end points and the subscript denotes what coordinate to use for an end...

","url":"https://quantum-equations.com/equation-list/f/the-integral-f-rotated-about-either-axis-is-equal","title":"The Integral f Rotated About either Axis is Equal ","summary":"Note: f(x) rotated about the x-axis between x = a_x and x = b_x, and f(x) rotated about the y-axis between y = a_y and b_y. Where a and b are the end points and the subscript denotes what coordinate to use for an end...","date_modified":"2026-04-04T19:08:12Z"},{"id":"https://quantum-equations.com/equation-list/f/integral-equals-difference-of-ordinates-at-a-and-b","html_content":"

Equation: y = ce^x

","url":"https://quantum-equations.com/equation-list/f/integral-equals-difference-of-ordinates-at-a-and-b","title":"Integral Equals difference of Ordinates at a and b","summary":"Equation: y = ce^x","date_modified":"2026-04-03T22:55:48Z"},{"id":"https://quantum-equations.com/equation-list/f/arc-length-equals-the-difference-between-end-points-x%5E2-2","html_content":"

There are different ways to express this equation with log or ln. The use of ln will return subscripts for the difference ((x(B)^2 / 2) - (x(A)^2 / 2)), while log will return the equation x(B)^2 / 2

","url":"https://quantum-equations.com/equation-list/f/arc-length-equals-the-difference-between-end-points-x%5E2-2","title":"Arc Length Equals the Difference between End Points x^2 / 2","summary":"There are different ways to express this equation with log or ln. The use of ln will return subscripts for the difference ((x(B)^2 / 2) - (x(A)^2 / 2)), while log will return the equation x(B)^2 / 2","date_modified":"2026-04-03T17:05:12Z"},{"id":"https://quantum-equations.com/equation-list/f/normal-is-bisected-by-f-at-p-and-coordinate-axis","html_content":"

Equation: c = x^2 - y^2

","url":"https://quantum-equations.com/equation-list/f/normal-is-bisected-by-f-at-p-and-coordinate-axis","title":"Normal is Bisected by f at P and Coordinate axis","summary":"Equation: c = x^2 - y^2","date_modified":"2026-04-03T16:29:41Z"},{"id":"https://quantum-equations.com/equation-list/f/equation-bisector-from-origin","html_content":"

Equation: y = ((y(A) * y') / (2 * x(A) * y' - y(A))) * (x - (x(A) - (y(A) / 2y'))) + (y(A) / 2)

","url":"https://quantum-equations.com/equation-list/f/equation-bisector-from-origin","title":"Equation Bisector from Origin","summary":"Equation: y = ((y(A) * y') / (2 * x(A) * y' - y(A))) * (x - (x(A) - (y(A) / 2y'))) + (y(A) / 2)","date_modified":"2026-04-03T15:45:05Z"},{"id":"https://quantum-equations.com/equation-list/f/tangent-at-p-line-to-origin-and-x-axis-make-isosceles","html_content":"

Equation: y = c / (k * x)

","url":"https://quantum-equations.com/equation-list/f/tangent-at-p-line-to-origin-and-x-axis-make-isosceles","title":"Tangent at P, Line to Origin, and x-axis make Isosceles","summary":"Equation: y = c / (k * x)","date_modified":"2026-04-03T14:39:21Z"},{"id":"https://quantum-equations.com/equation-list/f/p-of-tangent-bisected-by-coordinate-axis-and-intercepts","html_content":"

Equation: y = (c / x)

","url":"https://quantum-equations.com/equation-list/f/p-of-tangent-bisected-by-coordinate-axis-and-intercepts","title":"P of Tangent Bisected by Coordinate axis and Intercepts","summary":"Equation: y = (c / x)","date_modified":"2026-04-03T02:27:00Z"},{"id":"https://quantum-equations.com/equation-list/f/isosceles-from-normal-a-line-from-p-to-00-and-x-axis","html_content":"

Equation: c = x^2 - y^2

","url":"https://quantum-equations.com/equation-list/f/isosceles-from-normal-a-line-from-p-to-00-and-x-axis","title":"Isosceles from Normal, a line from P to (0,0), and x-axis","summary":"Equation: c = x^2 - y^2","date_modified":"2026-04-03T01:54:27Z"},{"id":"https://quantum-equations.com/equation-list/f/tangent-x-intercept-equals-x-coordinate-at-p","html_content":"

Equation: y = ( y^2 + x^2) / c

","url":"https://quantum-equations.com/equation-list/f/tangent-x-intercept-equals-x-coordinate-at-p","title":"Tangent x intercept Equals Length From P to x Intercept","summary":"Equation: y = ( y^2 + x^2) / c","date_modified":"2026-04-03T00:59:35Z"},{"id":"https://quantum-equations.com/equation-list/f/subtangent-equals-sum-of-x-and-y-coordinates","html_content":"

Equation: x = y * ln| c * y |

","url":"https://quantum-equations.com/equation-list/f/subtangent-equals-sum-of-x-and-y-coordinates","title":"Subtangent Equals Sum of x and y Coordinates","summary":"Equation: x = y * ln| c * y |","date_modified":"2026-04-03T00:25:12Z"},{"id":"https://quantum-equations.com/equation-list/f/slope-equals-square-of-abscissa","html_content":"

Equation: y = (x^3 / 3) + c

","url":"https://quantum-equations.com/equation-list/f/slope-equals-square-of-abscissa","title":"Slope Equals Square of Abscissa","summary":"Equation: y = (x^3 / 3) + c","date_modified":"2026-04-02T23:52:22Z"},{"id":"https://quantum-equations.com/equation-list/f/normal-between-p-and-x-intercept-is-constant","html_content":"

Equation: -(k^2 - y^2)^(1/2) = x + c

","url":"https://quantum-equations.com/equation-list/f/normal-between-p-and-x-intercept-is-constant","title":"Normal between P and x intercept is Constant","summary":"Equation: -(k^2 - y^2)^(1/2) = x + c","date_modified":"2026-04-02T20:49:21Z"},{"id":"https://quantum-equations.com/equation-list/f/tangent-from-p-to-x-intercept-is-constant","html_content":"

Equation: x = (k^2 - y^2)^(|1/2) - k * ln[(k + (k^2 - y^2)^(1/2)) / y] + c

","url":"https://quantum-equations.com/equation-list/f/tangent-from-p-to-x-intercept-is-constant","title":"Tangent from P to x intercept is Constant","summary":"Equation: x = (k^2 - y^2)^(|1/2) - k * ln[(k + (k^2 - y^2)^(1/2)) / y] + c","date_modified":"2026-04-02T20:20:42Z"},{"id":"https://quantum-equations.com/equation-list/f/tangent-x-intercept-equals-ordinate","html_content":"

Equation: x = (y * lny - (c * y))

","url":"https://quantum-equations.com/equation-list/f/tangent-x-intercept-equals-ordinate","title":"Tangent x-intercept Equals Ordinate","summary":"Equation: x = (y * lny - (c * y))","date_modified":"2026-04-01T21:06:47Z"},{"id":"https://quantum-equations.com/equation-list/f/normal-between-f-and-y-axis-bisected-by-the-x-axis","html_content":"

Equation: y^2 = (-x^2 / 2) + c

","url":"https://quantum-equations.com/equation-list/f/normal-between-f-and-y-axis-bisected-by-the-x-axis","title":"Normal Between f and y-axis Bisected by the X-axis","summary":"Equation: y^2 = (-x^2 / 2) + c","date_modified":"2026-04-01T18:04:52Z"},{"id":"https://quantum-equations.com/equation-list/f/subnormal-equals-the-square-of-x-coordinate","html_content":"

Equation: y^2 = ((2 * k * x^3) / 3) + c

","url":"https://quantum-equations.com/equation-list/f/subnormal-equals-the-square-of-x-coordinate","title":"Subnormal Equals the Square of X-coordinate","summary":"Equation: y^2 = ((2 * k * x^3) / 3) + c","date_modified":"2026-04-01T13:14:11Z"},{"id":"https://quantum-equations.com/equation-list/f/subnormal-equals-constant-k","html_content":"

Equation: y^2 = 2 * k * x + 2 * k * c ;

","url":"https://quantum-equations.com/equation-list/f/subnormal-equals-constant-k","title":"Subnormal Equals Constant k","summary":"Equation: y^2 = 2 * k * x + 2 * k * c ;","date_modified":"2026-04-01T12:53:28Z"},{"id":"https://quantum-equations.com/equation-list/f/subtangent-equals-constant-k","html_content":"

Equation: y^k = c * e^x

","url":"https://quantum-equations.com/equation-list/f/subtangent-equals-constant-k","title":"Subtangent Equals constant k","summary":"Equation: y^k = c * e^x","date_modified":"2026-04-01T10:54:11Z"},{"id":"https://quantum-equations.com/equation-list/f/tangent-equals-sum-of-coordinates-at-pxy","html_content":"

Equation: c * e^x - x - 1

","url":"https://quantum-equations.com/equation-list/f/tangent-equals-sum-of-coordinates-at-pxy","title":"Tangent Equals Sum of Coordinates at P(x,y)","summary":"Equation: c * e^x - x - 1","date_modified":"2026-04-01T10:23:48Z"}]}