Suppose Guido is romping on the surface \(z=x^2+y^2.\)
(a)
Suppose his path (or path’s shadow in the \(xy\)-plane) is \(x=t, y=1\text{.}\)
(i)
What is the equation \(P(t)\) of the space curve he is following?
(ii)
If he never steps off this path, for all practical purposes is Guido romping on a surface or on a curve?
(iii)
Calculate the instantaneous rate of change of his altitude (\(z\) from \(P(t)\)) at \(x=5,\)\(x=-3,\) and \(x\text{.}\)
(b)
Suppose his path is \(x=t, y=4\text{.}\)
(i)
What is the equation \(P(t)\) of the space curve he is following?
(ii)
Calculate the instantaneous rate of change of his altitude (\(z\)) at \(x=5,\)\(x=-3,\) and \(x.\)
(c)
Suppose his path is \(x=-3, y=t\text{.}\)
(i)
What is the equation \(P(t)\) of the space curve he is following?
(ii)
Calculate the instantaneous rate of change of his altitude (\(z\)) at \(y=4,\)\(y=-2,\) and \(t.\)
(d)
Suppose his path is \(x=7, y=t\text{.}\)
(i)
What is the equation \(P(t)\) of the space curve he is following?
(ii)
Calculate the instantaneous rate of change of his altitude (\(z\)) at \(y=4,\)\(y=-2,\) and \(t.\)
(e)
If Guido is moving parallel to an axis, in how many dimensions does the rate of change problem exist?
(f)
How many variables are in use as a result?
(g)
How does this help us use single variable calculus to solve this multivariable problem?
Subsection2.2.2Notation and Calculation
We call derivatives with respect to one or more variables of a multi-variable function partial derivatives. Partial Derivative Notations shows common notations.
List2.2.1.Partial Derivative Notations
First derivatives
\(\frac{\partial f}{\partial x}\) is the derivative of \(f\) with respect to \(x\) (which means \(y\) is held constant, i.e., \(y=k\)).
\(\frac{\partial f}{\partial y}\) is the derivative of \(f\) with respect to \(y\) (which means \(x\) is held constant, i.e., \(x=k\)).
Higher derivatives
\(\frac{\partial^2 f}{\partial x \partial y}\) is the derivative of \(\frac{\partial f}{\partial y}\) with respect to \(x\) (i.e., first \(y\) then \(x\)).
\(\frac{\partial^2 f}{\partial x^2}\) is the derivative with respect to \(x\) both times.
\(\frac{\partial^3 f}{\partial y \partial x \partial y}\) with respect to \(y\) then \(x\) then \(y\) again.
Note a common alternate notation uses the reversed notation \(f_{xy}=\frac{\partial^2 f}{\partial y \partial x}\text{.}\)
Compare \(\frac{\partial^2 f}{\partial x \partial y}\) and \(\frac{\partial^2 f}{\partial y \partial x}\text{.}\)
The following theorem can be proved.
Theorem2.2.3.Clairaut.
If \(f\) is defined on all points \(P\) such that \(\|P-C\| \le r\) for some point \(C\) and some number \(r\) and \(\frac{\partial^2 f}{\partial x \partial y}\) and \(\frac{\partial^2 f}{\partial y \partial x} \) are continuous, then at the point \(C,\)\(\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}. \)
