Suppose that f is a function of two variables (y and z) only. Show that the gradient transforms as a vector under rotations, Eq 1.29. [Hint: and the analogous formula for . We know that and ”solve” these equations for y and z (as functions of and (as functions of and ), and compute the needed derivatives , etc]
The matrix proves that’s is used to transform the vector under the rotation.
Consider the change in coordinates is .
Here, the variables of the function are y and z. The shifted coordinates are and are the changed coordinates. The angle of rotation is .
Rewrite the equations as,
Add the equation for the two coordinates as,
Rewrite the coordinates as,
Subtract the two equations as,
Determine the partial derivatives of the equation (1).
Determine the partial derivatives of the equation (2).
Write the expression for the gradient of the function with respect to y.
Write the equation for the gradient in terms of the partial derivative as,
Write the expression for the gradient in terms of the z.
Write the expression for the gradient in the matrix form.
Thus, the matrix proves that’s the is used to transform the vector under the rotation.
The height of a certain hill (in feet) is given by
Where y is the distance (in miles) north, x the distance east of South Hadley.
(a) Where is the top of hill located?
(b) How high is the hill?
(c) How steep is the slope (in feet per mile) at a point 1 mile north and one mileeast of South Hadley? In what direction is the slope steepest, at that point?
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