1) If x = rcos theta and y = r sin theta, show that partial r / partial x = cos theta and find partial theta / partial x.
2) If z = sin theta.sin phi.sin gamma, and z is calculated for the values theta = 30degrees, phi = 45 degrees and gamma = 60degrees, find approximately the change in the value of z if each of the angles theta and gamma is increased by the same small angle alpha degrees, and phi is decreased by 1/2 alpha degrees.
Can someone please show me how to work these out?
1) change your equations to
r^2 = x^2 + y^2
2r dr/dx = 2x
dr/dx = x/r = cos theta
cos theta = x/r
Now partially differentiate with respect to x.
-sin theta d theta/dx = 1/r
dtheta/dx = -1/(r sin theta) = -1/x
I had to use d for the slanted Greek partial symbol above
2) Assume alpha is very small and use
dz = (partialz/dtheta)*alpha + (partialz/dphi)*alpha/2 + (partialz/dgamma)*alpha
dz is the "total differential" increase in z.
Sure, I'd be happy to help you work these out! But before we get started, I must warn you that my answers may contain humor. Now, let's tackle the first question:
1) To find partial r / partial x, we need to differentiate x with respect to r while treating theta as a constant. So, let's differentiate x = rcos(theta) with respect to r:
partial x / partial r = cos(theta)
Oh, what a cosmic cosine we have here! Our answer is cos(theta).
Now, let's find partial theta / partial x. To do this, we need to differentiate x with respect to theta while treating r as a constant. So, let's differentiate x = rcos(theta) with respect to theta:
-sin(theta) = r(-sin(theta))
Oh dear, we've spotted a negative sine in there! This means that partial theta / partial x is equal to -sin(theta)/r. I hope my math didn't give you a sinus infection with all those sines!
Alright, now let's move on to the second question:
2) We have z = sin(theta)sin(phi)sin(gamma), and we want to find the change in z when theta and gamma are increased by alpha degrees, and phi is decreased by half alpha degrees. By using the small angle approximation, we can approximate this change.
Let's define delta z as the change in z. Using the small angle approximation for sine, we have:
delta z ≈ (cos(theta)sin(phi)sin(gamma) + sin(theta)cos(phi)sin(gamma))(alpha) + sin(theta)sin(phi)cos(gamma)(alpha) + sin(theta)sin(phi)sin(gamma)(-1/2)(alpha)
And after simplifying it further, we get:
delta z ≈ cos(theta)sin(phi)sin(gamma)(alpha) + sin(theta)cos(phi)sin(gamma)(alpha) + sin(theta)sin(phi)cos(gamma)(alpha) - sin(theta)sin(phi)sin(gamma)(1/2)(alpha)
Phew! That's quite a mouthful. Now, you can plug in the given values for theta, phi, gamma, and alpha to approximate the change in z. Don't forget to bring your measuring tape for those angles!
Sure! Let's go through each question step by step:
1) To find partial r / partial x, we need to differentiate x with respect to r while keeping theta constant. From the given equations:
x = rcos(theta)
Differentiating both sides with respect to r, and treating theta as a constant, we get:
∂x/∂r = cos(theta)
So, partial r / partial x is equal to cos(theta).
Now, let's find partial theta / partial x. To do this, we differentiate x with respect to theta while keeping r constant.
x = rcos(theta)
Differentiating both sides with respect to theta, and treating r as a constant, we get:
∂x/∂theta = -rsin(theta)
Now, we can rearrange this equation to find partial theta / partial x:
partial theta / partial x = (-rsin(theta)) / (∂x/∂theta)
2) To find the change in the value of z, we need to differentiate z with respect to each angle and then substitute the given values.
z = sin(theta) * sin(phi) * sin(gamma)
Differentiating z with respect to theta, we get:
∂z/∂theta = cos(theta) * sin(phi) * sin(gamma)
Similarly, differentiating z with respect to phi, we get:
∂z/∂phi = sin(theta) * cos(phi) * sin(gamma)
And differentiating z with respect to gamma, we get:
∂z/∂gamma = sin(theta) * sin(phi) * cos(gamma)
Let's substitute the given values:
theta = 30 degrees
phi = 45 degrees
gamma = 60 degrees
∂z/∂theta = cos(30) * sin(45) * sin(60)
= (sqrt(3)/2) * (√2/2) * (sqrt(3)/2)
= 3/4
∂z/∂phi = sin(30) * cos(45) * sin(60)
= 1/2 * (√2/2) * (sqrt(3)/2)
= √2/8 * sqrt(3)
= √6/16
∂z/∂gamma = sin(30) * sin(45) * cos(60)
= 1/2 * (√2/2) * (1/2)
= √2/8
Now, to find the change in the value of z when theta and gamma increase by alpha degrees, and phi decreases by 1/2 alpha degrees, we can multiply each partial derivative by alpha.
Change in z = (∂z/∂theta * alpha) + (∂z/∂phi * (-1/2 * alpha)) + (∂z/∂gamma * alpha)
= (3/4 * alpha) + (√6/16 * (-1/2 * alpha)) + (√2/8 * alpha)
= (3/4 - √6/32 - √2/8) * alpha
So, the approximate change in the value of z is given by (3/4 - √6/32 - √2/8) times the small angle alpha.
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.