Sure! Let's work through each problem step by step.
1) To show that partial r / partial x = cos theta, we need to take the partial derivative of r with respect to x.
Given x = rcos theta, we can solve for r in terms of x and theta: r = x / cos theta.
Now, we can find the partial derivative of r with respect to x:
partial r / partial x = 1 / cos theta * partial(x) / partial x
Since partial(x) / partial x = 1, we have:
partial r / partial x = 1 / cos theta
Therefore, partial r / partial x = cos theta.
To find partial theta / partial x, we need to find the partial derivative of theta with respect to x.
From x = rcos theta, we can solve for theta in terms of x and r: theta = arccos(x / r).
Now, we can find the partial derivative of theta with respect to x:
partial theta / partial x = partial(arccos(x / r)) / partial x
Using the chain rule, this can be simplified as:
partial theta / partial x = -1 / (sqrt(1 - (x / r)^2)) * (1 / r) * partial(r) / partial(x)
We already know that partial r / partial x = cos theta, and we can substitute this value into the equation:
partial theta / partial x = -1 / (sqrt(1 - (x / r)^2)) * (1 / r) * cos theta
Therefore, partial theta / partial x = -cos theta / (r * sqrt(1 - (x / r)^2)).
2) To find the change in the value of z when angles theta and gamma are increased by alpha degrees and phi is decreased by 1/2 alpha degrees, we need to calculate the partial derivatives of z with respect to each angle and multiply them by the corresponding change in each angle.
Given z = sin theta * sin phi * sin gamma, we can find the partial derivatives:
partial z / partial theta = cos theta * sin phi * sin gamma
partial z / partial phi = sin theta * cos phi * sin gamma
partial z / partial gamma = sin theta * sin phi * cos gamma
Now, let's calculate the change in z:
delta z = (partial z / partial theta) * (delta theta) + (partial z / partial phi) * (delta phi) + (partial z / partial gamma) * (delta gamma)
Since each angle is increased by alpha, we can substitute delta theta = delta phi = delta gamma = alpha into the equation:
delta z = (partial z / partial theta) * alpha + (partial z / partial phi) * (-1/2 * alpha) + (partial z / partial gamma) * alpha
Substituting the given values theta = 30 degrees, phi = 45 degrees, gamma = 60 degrees, and evaluating the partial derivatives, we get:
delta z = (cos 30 * sin 45 * sin 60) * alpha + (sin 30 * cos 45 * sin 60) * (-1/2 * alpha) + (sin 30 * sin 45 * cos 60) * alpha
Simplifying this expression will give you the approximate change in the value of z when the angles are changed by alpha degrees.