Chestermiller said:Let this be the derivative of (A(θ)aθ(θ)+B(θ)ar(θ)) with respect to θ. Solve for A and B.
Chet
jbunniii said:Can you express the unit vector in terms of ##\theta##?
Take the derivative of the expression I gave with respect to θ, and set it equal to the function you are trying to integrate with respect to θ. Make use of the condition that the derivative of the unit vector in the θ direction with respect to θ is equal to minus the unit vector in the r direction, and the derivative of the unit vector in the r direction with respect to θ is equal to plus the unit vector in the θ direction. You should get two ordinary differential equations in the two unknowns A and B. Solve these equations for A and B.ForTheGreater said:I don't get it.
ForTheGreater said:I have an integral of aΘ cos(Θ) dΘ
a is the unit vector for Θ.
I'm not sure what to do with it in the integration. I know the unit vector equals a/abs(a) but that would give a mess of an integral cause of the abs(a).
I like the method you are suggesting here much better than the approach I suggested in post #6. It is much more straightforward.pasmith said:You need to express [itex]\mathbf{a}_\theta[/itex] in cartesian components in order to integrate it:
[tex]
\mathbf{a}_\theta = \frac1r \left(\frac{\partial x}{\partial \theta}\mathbf{e}_x +
\frac{\partial y}{\partial \theta}\mathbf{e}_y + \frac{\partial z}{\partial \theta}\mathbf{e}_z
\right) = \dots?
[/tex]
Actually, it should be [itex]-sin(\theta)\vec{i}+ cos(\theta)\vec{j}[/itex]. The relation you gave is for the unit vector in the radial direction (assuming θ is the angle measured counterclockwise from the x axis).HallsofIvy said:Is this in three dimensions or two? If it is in three dimensions, what angle is [itex]\theta[/itex]? If it is in two dimensions, then the unit vector in direction [itex]\theta[/itex] can be written [itex]cos(\theta)\vec{i}+ sin(\theta)\vec{j}[/itex].
Integration is a mathematical concept that involves finding the area under a curve or the accumulation of a quantity over a certain interval. It is the reverse process of differentiation, and is often used to solve problems in physics, engineering, and other fields.
Unit vectors are vectors that have a magnitude of 1 and are used to represent directions or coordinate axes in mathematical calculations. They are often denoted by the letters i, j, and k in three-dimensional space, and their components can be found using trigonometric functions.
Unit vectors are used in integration to represent the direction of a curve or the orientation of a surface. They can also be used to find the components of a vector that is being integrated, making the calculation easier and more accurate.
Definite integration involves finding the exact value of an integral over a specific interval, while indefinite integration involves finding a general solution that includes a constant of integration. Definite integration is often used to find the area under a curve, while indefinite integration is used to solve differential equations.
Integration has many applications in science, including finding the velocity and acceleration of an object in physics, determining the probability of events in statistics, and calculating the electric field and potential in electromagnetism. It is also used in many engineering fields, such as structural analysis and fluid dynamics.