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UNIVERSITY OF BRISTOL. School of Mathematics. MULTIVARIABLE CALCULUS ... This paper contains TWO questions. ... relative weighting of the questions.
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School of Mathematics
(Paper code MATH–20901J)
January 2017 1 hour 30 minutes
This paper contains TWO questions. Both answers are used for assessment.
Calculators are not permitted in this examination.
On this examination, the marking scheme is indicative and is intended only as a guide to the relative weighting of the questions.
Do not turn over until instructed.
Cont... MVC
F(x) ≈ (−^14 π^2 + πx, 12 π − x + y).
(iii) (8 marks) Giving reasons, state whether the pair of simultaneous equations u = x^2 + y^3 , v = cos x + sin y can be inverted to give (x, y) uniquely in terms of (u, v) in the neighbourhood of points (x 0 , y 0 ) lying, firstly, on the line x = 0 and, secondly, on the line x = 12 π.
(b) Now let G(r) =
−y x^2 + y^2
x x^2 + y^2
with r = (x, y, z).
(i) (10 marks) Calculate
C
G · dr as a path integral where C is the circle of radius a lying in the (x, y)-plane, centred on the origin and oriented anti-clockwise. (ii) (10 marks) State Green’s theorem in the plane and use it to calculate
C
G · dr as an integral over the area of the circle with boundary C. (iii) (6 marks) Do your two answers to parts (b)(i) and (b)(ii) coincide? If not, provide an explanation as to why they don’t.
Continued...
