Changes

for a curve in a volume.

for a curve in a volume.

Let J (not really the Jacobian, but we treat it that way) be the [[norm]] of that vector, that is, the square root of the sum of the squares of its 2 or 3 components.

Let J (not really the Jacobian, but we treat it that way) be the [[norm]] of that vector, that is, the square root of the sum of the squares of its 2 or 3 components.

::<math>J = ||\left(\frac{\partial x}{\partial t}, \frac{\partial y}{\partial t}, \frac{\partial z}{\partial t}\right)|| = \sqrt{\left(\frac{\partial x}{\partial t}\right)^2 + \left(\frac{\partial y}{\partial t}\right)^2 + \left(\frac{\partial z}{\partial t}\right)^2}</math>

::<math>J = \Bigg\Vert\left(\frac{\partial x}{\partial t}, \frac{\partial y}{\partial t}, \frac{\partial z}{\partial t}\right)\Bigg\Vert = \sqrt{\left(\frac{\partial x}{\partial t}\right)^2 + \left(\frac{\partial y}{\partial t}\right)^2 + \left(\frac{\partial z}{\partial t}\right)^2}</math>

Then

Then

::<math>\int f(t)\ J dt</math>

::<math>\int f(t)\ J dt</math>