Determine if the following statements are true or false. (a) if ∫c vector f · d vector r = 0 for one closed path c, then vector f is path-independent. true false (b) if the total change of a function f along a curve c is zero, then c must be a contour of f. true false (c) line integralc vector f · d vector r = vector f(q) - vector f(p) when p and q are the endpoints of c. true false (d) if vector f is continuous and vector f = gradf, then vector f is path-independent. true false (e) if f(x) and g(y) are continuous one-variable functions, then the vector field vector f = f(x)vector i + g(y)vector j is path-independent. true false (f) the vector field vector f( vector r ) = vector r in 3-space is path-independent.

step-by-step explanation:

given that there are two triangles j'k'l' and jkl>

the former is got by dilating the second one.

when drawn on a grid, (graph paper) coordinates of vertices of first triangle is

(6,12)j, (9,6)k and (12,9)l

side jk has length = distance between j and k

=

side j'k'=distance between j' and k'

=

ie j'k'l' has sides 1/3 of that of corresponding sides of jkl.

hence scale factor of 1/3 is used to get j'k'l'from jkl

answer 1 : option "c"

answer 2 : option "d"

answer 3 : option "b"

answer 4 : option "d"

answer 5 : option "b"

answer 6 : option "a"

answer 7 : option "c"

answer 8 : option "d"

answer 9 : option "c"

answer 10 : option "d"

answer 11 : option "c"

answer 12 option "b"

answer 13 option "c"

answer 14 option "c"

answer 15 option "d"

answer 16 option "a"

answer 17 option "b"

answer 18 option "c"

answer 19 option "a"

answer 20 option "a"

answer 21 option "c"

d

0.85% : )

step-by-step explanation: