(Original post by **Icarus**)

Find the work done by the force field F(x,y) = x sin j + yj on particle that moves along the parabola y = x² from (-1,1) to (2,4).

You need to know about line integrals of vector fields.

See

http://farside.ph.utexas.edu/~rfitzp...es/node13.html and

http://farside.ph.utexas.edu/~rfitzp...es/node14.html for good explanations of line integrals of vector fields.

(Original post by **Icarus**)

A 170lb man carries a 55 lb can of paint up a helical staircase that encircles a silo with a radius of 20ft. If the silo is 100 ft high and the man makes exactly three complete revolutions, how much work is done by the man against gravity in climbing to the top?

This is a simple mass*gravity_strength*vertical_h eight question, the helix stuff is irrelevant, as gravity operates vertically.

(Original post by **Icarus**)

Find the area enclosed by the ellipse x²/a² + y²/b² = 1

Write the ellipse in parametric form (x = a*cos(t), y = b*sin(t)) and then integrate. You should find yourself integrating 4*a*b*sin^2(t) between 0 and pi/2 with respect to t. You should end up with the area as a*b*pi.

(Original post by **Icarus**)

Find the flux of the vector field F(x,y,z) = z i + y i + x k across the unit sphere x² + y² + z² = 1.

You need to know about surface integrals of vector fields.

See

http://oregonstate.edu/dept/math/Cal...flux/flux.html,

http://farside.ph.utexas.edu/~rfitzp...es/node15.html and

http://farside.ph.utexas.edu/~rfitzp...es/node16.html for good explanations of flux in relation to surface integrals of vector fields.

(Original post by **Icarus**)

The temperature at a point in a ball with conductivity K is inversely proportional to the distance from the center of the ball. Find the rate of heat flow across a sphere S of a radius a with center at the center of the ball.

I think here you want surface integrals again (see the last question).

(Original post by **Icarus**)

Find the orthogonal trajectories of the family of curves for x² - y² = k and y = kx³

Given a family of curves, an orthogonal trajectory is a curve which intersects each member of the family at right angles (perpendicularly).

Recall for a curve to be perpendicular to another curve, their gradients must be the negative reciprocal of each other.

So for the curve family, x^2 - y^2 = k, let's differentiate:

2x - 2yy' = 0

y' = x/y

So for the orthogonal trajectory (negative reciprocals of gradients):

y' = -y/x

1/y dy = -1/x dx

ln|y| = - ln|x| + C

|y| = e^(- ln|x| + C) = A/(|x|)

So |xy| = A, where A > 0

thus xy = B, for any B, is the orthogonal trajectory family of curves.

You should be able to see how this works now.

(Original post by **Icarus**)

Solve the initial value problem for 1 + y cos xy + (x cos xy)y´ = 0, y(2) = 0

Never heard of the initial value problem, but I assume it just means to solve the above differential equation for the particular case when x = 2 and y = 0.

(Original post by **Icarus**)

Solve the equation y´´ - 6y´ + 20y = 0

A standard second-order differential equation as in P4 A-level maths.

(Original post by **Icarus**)

Show that there is no vector field G such that

curl G = 2xi + 3yzj – xz²k

You need to read up on vector fields and things like curl and divergence. See

http://farside.ph.utexas.edu/~rfitzp...res/node4.html for a nice vector calculus tutorial.

(Original post by **Icarus**)

A 20.0 lb piece of hot copper is dropped into 30.0 lb of water at 60 degree F (Fahrenheit). If the final temperature of the mixture is 85 degree F, what was the initial temperature of the copper?

Find a physics text book somewhere!

(Original post by **Icarus**)

For what values of x does the graph of f(x) = 2x³ – 4x² – 9x + 100 have a horizontal tangent?

Look for when dy/dx = 0.

(Original post by **Icarus**)

A lighthouse is on a small island 8km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1.2km from P?

Draw a little diagram and use v = r*omega circular motion equation along with some basic trigonometry...

(Original post by **Icarus**)

Find all functions g such that

g´(x) = 8 sin x – 6x^5 + 6x^¾

Basic, A-level integration.

(Original post by **Icarus**)

A wire takes the shape of the semicircle x² + y² = 1, y ≥ 0, and is thicker near its base than near the top. Find the center of mass of the wire if the linear density at any point is proportional to its distance from the line y = 1.

I believe this sort of stuff is covered in M3 centre of mass (maybe without varying density, but that shouldn't be too hard to account for), so try and get hold of an M3 textbook...

(Original post by **Icarus**)

Now, obviously I don't want you to answer any or all of the questions for me (although that would be nice)... but because I don't recall doing any of this for A Level Maths (except maybe the derivatives and functions) can you tell me which areas of Maths I need to research into? Just short headings for each questions would be nice.

I hope this has helped you some. Who asked you these questions? They seem a bit difficult for someone with no experience in the areas they involve, as a few of them are way beyond A-level maths and further maths.

Well, good luck!

Regards,