Homework Statement
1.
(a) Remove all the internal bubbles in the circuit by applying DeMorgan's theorem so that the circuit consists of only AND gates and OR gates, and INVERTERS. INVERTERS can only be used for the inversion of inputs. Note that there is a 3-input NAND gate shared by both F1...
Hmm interesting. I've got it now. The electric potential inside the shell is the sum of the electric potential on the outer edge and the inner edge.
This yields: V=k\frac{3Q}{4r_1}
Ok, but I'm still confused on the next question. I would think the electric potential inside the cavity would...
Yes, I know that
vec(E)=-vec(\nabla V)
I also know that
E=-\frac{dV}{ds} when ds||E
If I use the second equation I set E equal to zero and you have v=0 which is wrong. The first one I'm not even sure how to use since I have magnitudes of vectors not vectors.
Homework Statement
A hollow spherical conductor, carrying a net charge +Q, has inner radius r1 and outer r2=2r1 radius (see the figure ). At the center of the sphere is a point charge +Q/2.
A.Write the electric field strength E in all three regions as a function of r: for r>r2,r1<r<r2...
Homework Statement
I need to solve this DE system for a lab:
q_1'=2-\frac{6}{5}q_1+q_2
q_2'=3+\frac{3}{5}q_1-\frac{3}{2}q_2
Homework Equations
The Attempt at a Solution
I know how to use the method of elimination to solve such systems, but this is non homogeneous because of the added...
you can rewrite the exponents as (x-y)(x+y). Using the substitution u=x-y and y=x+y you have a parallelogram with bounds v=3u,v=3u-8,v=-2u+1,v=-2u+8. Therefore, the area can be represented by the following integrals...
Homework Statement
Evaulate the integral making an appropriate change of variables.
\int\int_R(x+y)e^{x^2-y^2}dA where R is the parallelogram enclosed by the lines x-2y=0, x-2y=4, 3x-y=1, 3x-y=8 .
Homework Equations
The Attempt at a Solution
I'm not sure what change of variables I should...
Hmm, that's what I thought you were thinking. I handed in the assignment today and got the answer key. They setup the integral the way you did in the answer key. However, I would argue that part of the region in that integral is not under the cone. Everything under the cone is contained within a...
Ok, I think I was misinterperting the volume described. I took it to mean basically the shadow cast down by the cone on the xy-plane, since that region lies under the cone, above the xy-plane, and in the sphere. I'm not sure if the area you are thinking of is correct though. In the area you are...
Oh, haha. Cant believe I forgot that. I would think the upper limit on rho depends on phi because with this shape, you can change theta all you want and rho is not going to change. However, changing phi changes rho, this is because the solid is symmetrical about the z-axis. Right?
Homework Statement
Using spherical coordinates, find the volume of the solid that lies within the sphere x2+y2+z2=4, above the xy-plane and below the cone z=√(x2+y2)
Homework Equations
The Attempt at a Solution
This is what I have so far...
update, I proved everything, however I'm not sure If my proof for b is what they're asking for. I said that
\int_{-\infty}^{\infty}e^{-x^2}dx\int_{-\infty}^{\infty}e^{-y^2}dy=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dy dx
and since I had proved that the rhs of the...
Homework Statement
(a) we define the improper integral (over the entire plane R2)
I=\int\int_{R^2}e^{-(x^2+y^2)}dA=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dy dx=\lim_{a\rightarrow\infty}\int\int_{D_{a}} e^{-(x^2+y^2)} dA
where Da is the disk with radius a and center the...
Homework Statement
show that the curvature of a plane curve is \kappa=|\frac{d\phi}{ds}| where phi is the angle between T and i; that is, phi is the inclination of the tangent line.
Homework Equations
The Attempt at a Solution
I'm not sure how to start this one out.
Any ideas?
Homework Statement
A rocket burning it's onboard fuel while moving through space has a velocity v(t) and mass m(t) at time t. If the exhaust gasses escape with velocity ve relative to the rocket , it can be deduced from Newton's Second Law of Motion that...
Ok, I stared at the problem for a good amount of time and I think I have it. I managed to manipulate a few things...
2( \vec{r}(t) \cdot \vec{r}'(t))=\frac{d}{dt}(\vec{r}(t)\cdot\vec{r}(t))=\frac{d}{dt}(\left\|\vec{r}(t)\right\|^2)=0
so it must be true then that also...
Hmm.
Lineintegral1:
Ok, I can rewrite what I have written.
\vec{r}(t) \cdot \vec{r}'(t)=0
\left\|\vec{r}(t) \right\|\left\|\vec{r}'(t)\right\|cos\theta=0
||r(t)|| has to equal the radius of the sphere. However, isen't that what I'm suppose to be proving given that r(t) and r'(t) are...
Homework Statement
if a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t). show that the curve lies on a sphere with center the origin.
Homework Equations
The Attempt at a Solution
I'm not quite sure how to prove this.
I...
Homework Statement
Find an equation for the surface consisting of all points p for which the distance from P to the x-axis is twice the distance from P to the yz-plane. Identify the surface.
Homework Equations
The Attempt at a Solution...
Yes, but since the vectors have to be unit vectors and both vectors a and c are orthogonal to b, |v| is going to be the same regardless. You can rotate a vector by a specific angle and the others have to stay orthogonal so they will move as well by the angle maintaining |v|. That's why I chose...
Homework Statement
A.
Given unit vectors a, b, c in the x, y-plane such that a · b = b · c = 0,
let v = a + b + c; what are the possible values of |v|?
B.
Repeat, except a, b, and c are unit vectors in 3-space
Homework Equations
The Attempt at a Solution
I have solutions for both that I'm...
I can replace the c\bulletb side with what you've suggested but then how am I supposed to include that it is twice the angle. I loose this ability without the trigonometric function.
I'm not quite sure what you are suggesting.
I can draw the relationship
|a||b|cos \theta = |c||b| cos 2\theta
I can eliminate the |b| from both sides, but I don't know where to go from there, since |c| doesn't seem to help when substituting.
Homework Statement
Given a = <1,2,3> and b = <1,-1,-1>, sketch the collection of all position vectors c satisfying a x b = a x c
Homework Equations
The Attempt at a Solution
I've calculated a x b = <1,4,-3> and Defining c = <x,y,z> I found a x c = <2z-3y, z-3x, y-2x>. I want to...
Homework Statement
If
c=|a|b+|b|a where a,b, and c are all non zero vectors, show that c bisects the angle between a and b
Homework Equations
The Attempt at a Solution
I'm taking the approach to prove that the angle between b and c= the angle between c and a
I have written...