FunQcmen+al Taorem of Ca&culuS Vcx3j=T (9)) df dX = f(b)-fla) dx f(a)
7 b
I1o j9 wmS Afs from a fo b
f eva&uakI a =@ X=b minuS f evaIuskI @ X=Q
Fundamental
Theorem of Gra&;en+s (FTG)
(V).d=T(b) -Tla)
T
This is indepen&at of path
ExampL : Tes+ of FTG
Giua T=xy3
w1 a=(o,0,o) anI 5=(z,lO)
EUal RHS of+L FT6
1(67-T(a)=T(2,1,0)-T(00,0) =a-O=2
Eva&uak LHS of fu FT6
I y+d
(2,1,0)
e 2 un?fs tu Me righf dL s+idx x:0 +o2 y=o : l unit upward dij-+jdy x=2 y:o +o l
(0,0,0)
x
aT Ie>+ ay 2z
ax
x) e ax
2(xy3) + K 2_(xy3) 22
+ k(o)
1z P
+
2Xy
(o'1'z7 TP Z4J
(z,1,0) TP.L
TP.-Z
0.0,0
(0'.'0]
(117 4
49rees W1 IZHS T(z,1O)=T(o0,o)
47 Test ahotar path
Pa+h 2
'1z X
+
di=idx +jd