( 6 points ) Calculate the Laplacian of the following scalar fields a ) T x 2 2 x y 3 x 4 b ) T s t n ( t ) t i n ( y ) s i n ( t ) c ) T e 5 x s i n ( 4 y ) c o s ( 3 x ) ( 6 points ) Piove that the divergence of a curl and the curl of a grhdient are always zero ( 6 pointa ) Check the fundamental theorem for gradicnts, using T x 2 4 x y 2 y z 3 , the points vec ( d ) ( 0 , 0 , 0 ) , vec ( b ) ( 1 , 1 , 1 ) , and the two paths ( thown in the figure to the right as well ) a ) ( 0 , 0 , 0 ) ( 1 , 0 , 0 ) ( 1 , 1 , 0 ) ( 1 , 1 , 1 ) b ) ( 0 , 0 , 0 ) ( 0 , 0 , 1 ) ( 0 , 1 , 1 ) ( 1 , 1 , 1 ) ( 8 points ) For the vector field hat ( v ) xyI 2 y x j 3 , a ) Test the divergonce theorem for this vector field Take as your volume the cube shown belik ( let ) , with sides of length 2 b ) Teat 'Stokes ' theorem for this vector fleld, using the triangular shaded area in the figare a ) Compate the divergence of v b ) Checi the divergosee Dheorms for iv , using as your volumse De itverfed hemispherical boal of radius R , resting on the x y plase and centered at the origis, as shown to the right ( 8 points ) For the scalar field T r ( c o s s i n c o s ) , a ) Compute the gradient of T b ) Compute the Laplacian of T c ) Text the gradicnt theoecm for this function, using the path shown in the figure to the right, from ( 0 , 0 0 ) to ( 0 , 0 , 2 ) ( 8 points ) Theory of vector fields a ) Let vec ( F 1 ) x 2 f and vec ( F ) 2 x P y 9 x z Calculate the divergence and curl of these vector fields Which one can be written as the gradient of a scalar field Find a scalar potential that does the job Which one can be written as the curl of a vector field Find a suitable vector potential b ) Show that vec ( F ) yzi z x 9 x y ( 6 points ) Calculate the Laplacian of the following scalar fields a ) T x 2 2 x y 3 x 4 b ) T s t n ( t ) t i n ( y ) s i n ( t ) c ) T e 5 x s i n ( 4 y ) c o s ( 3 x ) ( 6 points ) Piove that the divergence of a curl and the curl of a grhdient are always zero ( 6 pointa ) Check the fundamental theorem for gradicnts, using T x 2 4 x y 2 y z 3 , the points vec ( d ) ( 0 , 0 , 0 ) , vec ( b ) ( 1 , 1 , 1 ) , and the two paths ( thown in the figure to the right as well ) a ) ( 0 , 0 , 0 ) ( 1 , 0 , 0 ) ( 1 , 1 , 0 ) ( 1 , 1 , 1 ) b ) ( 0 , 0 , 0 ) ( 0 , 0 , 1 ) ( 0 , 1 , 1 ) ( 1 , 1 , 1 ) ( 8 points ) For the vector field hat ( v ) xyI 2 y x j 3 , ( feel free to use results from 8 6 ) a ) Test the divergonce theorem for this vector field Take as your volume the cube shown belik ( let ) , with sides of length 2 b ) Teat 'Stokes ' theorem for this vector fleld, using the triangular shaded area in the figare a ) Compate the divergence of v b ) Checi the divergosee Dheorms for iv , using as your volumse De itverfed hemispherical boal of radius R , resting on the x y plase and centered at the origis, as shown to the right ( 8 points ) For the scalar field T r ( c o s s i n c o s ) , a ) Compute the gradient of T b ) Compute the Laplacian of T c ) Text the gradicnt theoecm for this function, using the path shown in the figure to the right, from ( 0 , 0 0 ) to ( 0 , 0 , 2 ) ( 8 points ) Theory of vector fields a ) Let vec ( F 1 ) x 2 f and vec ( F ) 2 x P y 9 x z Calculate the divergence and curl of these vector fields Which one can be written as the gradient of a scalar field Find a scalar potential that does the job Which one can be written as the curl of a vector field Find a suitable vector potential b ) Show that vec ( F ) yzi z x 9 x y

Question

( 6 points ) Calculate the Laplacian of the following scalar fields  a ) T   x 2 2 x y 3 x 4 b ) T   s t n ( t ) t i n ( y ) s i n ( t ) c ) T   e   5 x s i n ( 4 y ) c o s ( 3 x ) ( 6 points ) Piove that the divergence of a curl and the curl of a grhdient are always zero  ( 6 pointa ) Check the fundamental theorem for gradicnts, using T   x 2 4 x y 2 y z 3 , the points vec ( d )   ( 0 , 0 , 0 ) , vec ( b )   ( 1 , 1 , 1 ) , and the two paths ( thown in the figure to the right as well ) a ) ( 0 , 0 , 0 ) ( 1 , 0 , 0 ) ( 1 , 1 , 0 ) ( 1 , 1 , 1 ) b ) ( 0 , 0 , 0 ) ( 0 , 0 , 1 ) ( 0 , 1 , 1 ) ( 1 , 1 , 1 ) ( 8 points ) For the vector field hat ( v )   xyI 2 y x j 3 , a ) Test the divergonce theorem for this vector field  Take as your volume the cube shown belik ( let ) , with sides of length 2   b ) Teat  'Stokes ' theorem for this vector fleld, using the triangular shaded area in the figare a ) Compate the divergence of v   b ) Checi the divergosee Dheorms for iv , using as your volumse De itverfed hemispherical boal of radius R , resting on the x y plase and centered at the origis, as shown to the right  ( 8 points ) For the scalar field T   r ( c o s s i n c o s ) , a ) Compute the gradient of T   b ) Compute the Laplacian of T   c ) Text the gradicnt theoecm for this function, using the path shown in the figure to the right, from ( 0 , 0   0 ) to ( 0 , 0 , 2 )   ( 8 points ) Theory of vector fields a ) Let vec ( F 1 )   x 2 f and vec ( F ) 2   x P y 9 x z   Calculate the divergence and curl of these vector fields  Which one can be written as the gradient of a scalar field  Find a scalar potential that does the job  Which one can be written as the curl of a vector field  Find a suitable vector potential  b ) Show that vec ( F )   yzi z x 9 x y ( 6 points ) Calculate the Laplacian of the following scalar fields  a ) T   x 2 2 x y 3 x 4 b ) T   s t n ( t ) t i n ( y ) s i n ( t ) c ) T   e   5 x s i n ( 4 y ) c o s ( 3 x ) ( 6 points ) Piove that the divergence of a curl and the curl of a grhdient are always zero  ( 6 pointa ) Check the fundamental theorem for gradicnts, using T   x 2 4 x y 2 y z 3 , the points vec ( d )   ( 0 , 0 , 0 ) , vec ( b )   ( 1 , 1 , 1 ) , and the two paths ( thown in the figure to the right as well ) a ) ( 0 , 0 , 0 ) ( 1 , 0 , 0 ) ( 1 , 1 , 0 ) ( 1 , 1 , 1 ) b ) ( 0 , 0 , 0 ) ( 0 , 0 , 1 ) ( 0 , 1 , 1 ) ( 1 , 1 , 1 ) ( 8 points ) For the vector field hat ( v )   xyI 2 y x j 3 , ( feel free to use results from 8 6 ) a ) Test the divergonce theorem for this vector field  Take as your volume the cube shown belik ( let ) , with sides of length 2   b ) Teat  'Stokes ' theorem for this vector fleld, using the triangular shaded area in the figare a ) Compate the divergence of v   b ) Checi the divergosee Dheorms for iv , using as your volumse De itverfed hemispherical boal of radius R , resting on the x y plase and centered at the origis, as shown to the right  ( 8 points ) For the scalar field T   r ( c o s s i n c o s ) , a ) Compute the gradient of T   b ) Compute the Laplacian of T   c ) Text the gradicnt theoecm for this function, using the path shown in the figure to the right, from ( 0 , 0   0 ) to ( 0 , 0 , 2 )   ( 8 points ) Theory of vector fields a ) Let vec ( F 1 )   x 2 f and vec ( F ) 2   x P y 9 x z   Calculate the divergence and curl of these vector fields  Which one can be written as the gradient of a scalar field  Find a scalar potential that does the job  Which one can be written as the curl of a vector field  Find a suitable vector potential  b ) Show that vec ( F )   yzi z x 9 x y

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( 6 points ) Calculate the Laplacian of the following scalar fields: a ) T = x 2 2 x y 3 x 4 b

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