Exam Revision

Question 1

Find the equation of the tangent plane to the surface f(x, y, z) ≡ x^2+y^2+z^2=4 at the point (1, 1, 1)

Answer 1

The gradient of f is (2, 2, 4) at the point (x, y, z) = (1, 1, 1). The normal vector to the surface, and
therefore to the tangent plane is proportional to (2, 2, 4). The equation of the tangent plane
is of the form 2x+ 2y + 4z = K. Since the point (1, 1, 1) lies on the tangent plane we have
K = 2 + 2 + 4 = 8 and the equation of the plane is 2x + 2y + 4z = 8 or x + y + 2z = 4.

Question 2

Give the very useful formula

Answer 2

Question 3

Give the definition of differentiability for a scalar field

Answer 3

Question 4

What is the implicit function theorem

Answer 4

Question 5

Give the definition of a vector field being differentiable

Answer 5

Question 6

Give the definition of orientation preserving

Answer 6

Question 7

What is the differential

Answer 7

Question 8

Give the line integral formula

Answer 8

Question 9

Give method 1 for surface integrals

Answer 9

Question 10

Give method 2 for surface integrals

Answer 10

S is given as a level set of f

Question 11

What is Green’s Theorem in the plane

Answer 11

Question 12

Give the vector form of Green’s Theorem

Answer 12

Question 13

Give Stoke’s Theorem

Answer 13

Question 14

Give the divergence theorem

Answer 14

Question 15

Give the criteria for path independence

Answer 15

Question 16

Give Jacobian for a change of variables

Answer 16

Question 17

Give the formula for the dilation of a distribution

Answer 17

Question 18

Give the formula for multiplication of a distribution by a smooth factor

Answer 18

Question 19

Give the formla delta(f(x)) with f having simple zeros

Answer 19

Question 20

Give the definition of piecewise continuous

Answer 20

Question 21

Give the definition of piecewise smooth

Answer 21

Question 22

Give the formula for delta distribution under a change of co ordinates

Answer 22

Question 23

Give Green’s Formula

Answer 23

Question 24

Give the formula for the formal adjoint

Answer 24

Question 25

Give the formula for an SL operator

Answer 25

Question 26

Give the formula to relate an operator to an SL operator

Answer 26

Question 27

Give continuity requirement when solving Green's funcs

Answer 27

Question 28

Give jump discontinuity when solving Green's funcs

Answer 28

Question 29

Give the angle addition formulas

Answer 29

Question 30

Give the div operator in general form

Answer 30

Question 30

Give the curl operator in general form

Answer 30

Question 31

Give the grad operator in general form

Answer 31