1 consider a cubic room of side length a the vertices of which are 0 0
1. Consider a cubic room of side length a, the vertices of which are (0,0,0), (a,0,0),
(0, a, 0), (0, 0, a),(a, a, 0), (a, 0, a), (0, a, a), and (a, a, a).
At a certain moment, the temperature within this room is given by the scalar field
T(x, y, z)= Toxy z, where To is a constant.
(a) What are the physical dimensions of the constant To?
(b) Where is the hottest point in the room?
(c) Find the functions Tr-o(y, z), Tz-a (y, z), Ty=0(x, 2), Ty-a (x, z), T₂-0(x, y), Tz-a (x, y),
which give the temperature profile on each of the six walls of the room.
(d) In the centre of the room (i.e. the body-centre of the cube), what is the direction
of maximum increase of temperature? Comment on your answer./n2. Find the curl of the vector field A = x (xê, +yê, + zê₂).
Consider the vector field F = yêr+zêy + xêz.
(a) Calculate the divergence and the curl of this vector field.
Comment on your answers.
(b) Evaluate the line integral
where the contour C consists of: a straight line from the origin to (0, 0, b); a
quarter-circle centred at the origin from (0, 0, b) to (0, 6, 0); and a straight line
from (0, 6, 0) to the origin,
ii. using Stokes' Theorem.
Check that your answers agree.
/n4. Use the divergence theorem to evaluate the surface integral
ff (3x ê₂ + 2y êy) · ďªs,
where S is the surface of a sphere of radius 36 centred on the origin.
5. Find the divergence of the vector field A = x (zê₂ + xêy + yêz).
*The amount will be in form of wallet points that you can redeem to pay upto 10% of the price for any assignment. **Use of solution provided by us for unfair practice like cheating will result in action from our end which may include permanent termination of the defaulter’s account.Disclaimer:The website contains certain images which are not owned by the company/ website. Such images are used for indicative purposes only and is a third-party content. All credits go to its rightful owner including its copyright owner. It is also clarified that the use of any photograph on the website including the use of any photograph of any educational institute/ university is not intended to suggest any association, relationship, or sponsorship whatsoever between the company and the said educational institute/ university. Any such use is for representative purposes only and all intellectual property rights belong to the respective owners.