Verify Stoke's Theorem for Paraboloid-Z=9-x^2-y^2 above Plane-Z=5

[mohammadiqbal]

Hello everybody,
I am stuck on my homework. This question is giving me a headache, I've literally been staring at it for over 30 minutes. Perhaps someone here may be able to answer it (I hope), any help will be greatly appreciated:

Homework Statement

Legend:
S= integral
SS= double integral
sub= boundary/limit a
#^n= number raised to exponent n
*= multiply

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Let S be the portion of the paraboloid: z = 9 - x^2 - y^2 which lies above the plane: z= 5, with normal vector N pointing upward (i.e. in the direction of increasing z), and let F be the vector field given by: F(x,y,z)= (x-yz)i + xzj. Verify Stoke's Theorem by evaluation each of the following:a) The line integral: S sub c (F dr), where C is the boundary of s, oriented counter-clockwise relative to N.b) The flux integral: SS sub s (curl F) * N dS

Homework Equations

Stoke's Theorem
S sub c (F * dr) = S sub s S (curl F) * N ds

The Attempt at a Solution

I have no idea on how to start, I am soo lost :(

 

[mighty2000]

Hello,

I can definitely understand how this question can be confusing and overwhelming. Let's break it down step by step.

First, let's look at the given information. We have a paraboloid with an equation of z = 9 - x^2 - y^2. This means that the paraboloid is opening downwards and has a vertex at (0,0,9). The plane z = 5 intersects with the paraboloid, creating a circle with a radius of 2.

Next, we have a vector field given by F(x,y,z) = (x-yz)i + xzj. This means that the vector field has a component in the x direction and a component in the y direction, which are both dependent on the values of y and z. We will use this vector field to calculate the line integral and flux integral.

Now, let's look at the question itself. It is asking us to verify Stoke's Theorem by evaluating the line integral and flux integral. Stoke's Theorem states that the line integral along a closed curve is equal to the flux integral over the surface bounded by that curve. In this case, the curve is the boundary of the paraboloid, which is a circle with a radius of 2.

a) To evaluate the line integral, we will use the formula S sub c (F * dr), where C is the boundary of S. In this case, C is the circle with a radius of 2. Since the curve is oriented counter-clockwise relative to the upward normal vector, we will use a positive sign for the line integral. We can rewrite F as (x-yz)i + xzj = xi + (x-xz^2)j. Then, dr = dx + dy, since we are moving along the curve in the x and y directions. Plugging these values into the line integral formula, we get S sub c (F * dr) = S sub c (x dx + (x-xz^2) dy). We can now integrate this using the parametric equation of the circle, which is x = 2 cos(t) and y = 2 sin(t), where t ranges from 0 to 2π. This gives us S sub c (F * dr) = S sub 0 2π (2cos(t) (-2sin(t) dt) + (2cos(t) - 2cos(t)sin^