1. Find a scalar potential for the conservative vector field:u(x, y, z) = (zy sin(xy) + ye, zx sin(sy) + es, 1 — cos(xy)).2. Consider the vector fieldu(x, y, z) = sin(x), 2yz, y2). Calculate V • u and V x u. Is u irrotational ? Is u solenoidal ?3. Let f be a function and v be a vector field on Rft. Show thatV. (fv) = Of v + f(o v).4. Consider the functionf sz, z3) = xie" + x2 m*3). Calculate the second order Taylor expansion of f at a = (1,1,0). Include the remainder term R2(a, h). You don't have to say what the remainder term is.5. Consider the functionf(x,y,z) = 8x + 4y — 3z + xz + yz — 4x2 _ y2 z2. Find the one critical point of f and use the Hessian test to show that it is a local maximum.

1. Find a scalar potential for the conservative vector field: u(x, y, z) = (zy sin(xy) + ye, zx sin(sy) + es, 1 — cos(xy)). 2. Consider the vector field u(x, y, z) = sin(x), 2yz, y2). Calculate V • u...

1. Find a scalar potential for the conservative vector field:
u(x, y, z) = (zy sin(xy) + ye, zx sin(sy) + es, 1 — cos(xy)).
2. Consider the vector field
u(x, y, z) = sin(x), 2yz, y2). Calculate V • u and V x u. Is u irrotational ? Is u solenoidal ?
3. Let f be a function and v be a vector field on Rft. Show that
V. (fv) = Of v + f(o v).
4. Consider the function
f sz, z3) = xie" + x2 m*3). Calculate the second order Taylor expansion of f at a = (1,1,0). Include the remainder term R2(a, h). You don't have to say what the remainder term is.
5. Consider the function
f(x,y,z) = 8x + 4y — 3z + xz + yz — 4x2 _ y2 z2. Find the one critical point of f and use the Hessian test to show that it is a local maximum.

Answered Same DayDec 23, 2021

Answer To: 1. Find a scalar potential for the conservative vector field: u(x, y, z) = (zy sin(xy) + ye, zx...

Robert answered on Dec 23 2021

129 Votes

Sol:
Let the scalar potential be represented by ( ).
( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )
Substituting this in the other expressions, we get,
( ) ( )
( ) ( )
( )

( )
( ) ( ) ( ) ( )
( )
( ) ( )
Hence the scalar potential of the given vector field is given by,
( )

Sol:
( ) ( ( ) )
( ( )) ( ) (
) ( )
Hence the given field is not solenoidal.
|
̂ ̂ ̂

( )
|
Since the curl of the given field...

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1. Find a scalar potential for the conservative vector field: u(x, y, z) = (zy sin(xy) + ye, zx sin(sy) + es, 1 — cos(xy)). 2. Consider the vector field u(x, y, z) = sin(x), 2yz, y2). Calculate V • u...

Answer To: 1. Find a scalar potential for the conservative vector field: u(x, y, z) = (zy sin(xy) + ye, zx...

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