These questions relate to curvature problems.
a.Explain why (dÖ/ds) = (dÖ/dt) (dt/ds).
b.Explain why tanÖ = dy/dx, which equals (dy/dt) / (dx/dt).
I have to assume you know the chain rule.
if x is a function of t, and f is a function of x, then
df/dt=df/dx * dx/dt
this is a fundamental relationship in calculus.
I understand that much, but I still cannot derive the curvation equation..
a.Let x prime and x double prime be the first derivative and second derivatives of x with respect to t, and similarly for y. Show the following in true.
dƒ³/ds = ( x�Œ y�Œ�Œ - x�Œ�Œ y�Œ ) / (vector V)^3
Wow that did not work...
Prove
(dphi/ds) = ( xprime*ydouble prime - xdoubleprime*yprime ) / (vector V)^3
x prime. OK that is the first derivative of x with respect to ???
same for double prime. Your statement is just totally meaningless, it cant be proved unless it is understood.
I am trying to prove the formula..
a. To understand why (dÖ/ds) is equal to (dÖ/dt) multiplied by (dt/ds), we need to break down the differentials and understand their meanings.
Let's start with (dÖ/ds). Here, dÖ represents the change in the angle Ö, while ds represents the change in the arc length s. Essentially, (dÖ/ds) represents the rate of change of the angle Ö with respect to the arc length s.
Similarly, let's examine (dÖ/dt). Here, dÖ represents the change in the angle Ö, while dt represents the change in the parameter t. In this case, (dÖ/dt) represents the rate of change of the angle Ö with respect to the parameter t.
Now, let's consider dt/ds. Here, dt represents the change in the parameter t, while ds represents the change in the arc length s. dt/ds represents the rate of change of the parameter t with respect to the arc length s.
To understand why (dÖ/ds) = (dÖ/dt) multiplied by (dt/ds), we can recall the chain rule from calculus. The chain rule states that if a variable z is dependent on another variable y, which in turn depends on a variable x, then the rate of change of z with respect to x is given by the product of the rates of change of z with respect to y and y with respect to x.
In this case, Ö is dependent on t, which in turn depends on s. Therefore, the chain rule applies. By substituting the corresponding derivatives, we get (dÖ/ds) = (dÖ/dt) * (dt/ds), which explains the relationship between the differentials.
b. To understand why tanÖ is equal to dy/dx, which can be expressed as (dy/dt) / (dx/dt), we can again utilize the concepts from calculus.
Let's consider a curve in a coordinate system, where y represents the vertical axis, x represents the horizontal axis, and Ö represents the angle formed between the curve and the positive direction of the x-axis at any given point.
The tangent of an angle is defined as the ratio of the opposite side (dy) to the adjacent side (dx) in a right triangle formed by the angle. In this case, the opposite side represents the change in the y-coordinate (dy), and the adjacent side represents the change in the x-coordinate (dx).
Now, let's examine (dy/dx). Here, dy represents the change in the y-coordinate, while dx represents the change in the x-coordinate. (dy/dx) represents the rate of change of y with respect to x, which represents the slope of the curve.
Similarly, (dy/dt) represents the rate of change of y with respect to the parameter t, and (dx/dt) represents the rate of change of x with respect to t.
If we recall the definition of the tangent function, tanÖ = opposite/adjacent, we can substitute the corresponding derivatives and get tanÖ = dy/dx, which equals (dy/dt) / (dx/dt). This equation represents the relationship between the tangent of the angle formed by the curve and the rate of change of the y-coordinate with respect to the rate of change of the x-coordinate.