Question

Consider the vector field ????(x,y,z)=(5z+y)????+(4z+x)????+(4y+5x)????.

Answer 1

Answer:

a) $5xz + xy + 4yz$

b) 10

Step-by-step explanation:

a) Here $F(x,y,z)=(5z+y)i+(4z+x)j+(4y+5x)k$

Since the case $F$  =  ∇$f$ holds, then

∇$f = f_xi+f_yj+f_zk$ = $(5z+y)i+(4z+x)j+(4y+5x)k$

So, $f_x = 5z + y$

If we integrate $f_x$ with respect to x, we will get an integration constant C which is also a function that depends to y and z.

Hence,

$f = \int f_xdx = 5xz + xy + g(y,z)$

Now we need to find g(y,z).

So first let's take the derivative of g(y,z) with respect to y.

$f_y = x + g_y(y,z) = 4z + x$

Hence, $g_y(y,z) = 4z$

So now, if we integrate $g_y$ with respect to y to find g(y,z)

$g = \int g_ydy = 4yz + C$

Thus,

$f = 5xz + xy + g(y,z) = 5xz + xy + 4yz + C$

And since $f(0,0,0)=0$, then $C=0$

Thus,

$f = f(x,y,z) = 5xz + xy + 4yz$

b) By the Fundamental Theorem of Line Integrals, we know that

$\int\limits^a_b F. dr = F[r(b)]-F[r(a)]$

Hence,

$\int\limits^a_b F. dr = F(1,1,1)-F(0,0,0) =[(5+1+4)-(0+0+0)]=10$