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1. Find the gradient of the following functions at the given point and hence find the directional derivative in the direction of a given vector. a) T(x,y, z) = xi4y²+z®) cosx at (0, –1,3) along d = & – 39 + 2 2. b) T(x,y, z) = Vx² + z² + In xyz at (1,2,1) along d = 3 ŷ + 4 2 .

1. Find the gradient of the following functions at the given point and hence find the directional derivative in the direction of a given vector. a) T(x,y, z) = xi4y²+z®) cosx at (0, –1,3) along d = & – 39 + 2 2. b) T(x,y, z) = Vx² + z² + In xyz at (1,2,1) along d = 3 ŷ + 4 2 .

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Find the gradient of the following functions at the given point and hence find the directional derivative in the direction of a given vector. a) T(x,y,z) = cosx at (0, –1,3) along d = & – 3ŷ + 2 2. (x²+y²+z²) b) T(x,y,z) = Vx² +z² + In xyz at (1, 2, 1) along d = 3 ŷ +4 2. 2. Find the divergence and curl of the following functions a) A = & tan-1x +ŷ In sin y² + 2 sin z b) A = & cos²(x² +y³) +ŷ In (=) + 2 etan-z. 3. Find the Laplacian of the following functions a) T(x,y,z) = /x2 + y² + z² , b) T(x,y, z) = 3xy²+ y cos z + e. 4. Let A = & x?y – ŷ 3xy + 2 z², and consider the three-dimensional volume inside the cube with faces parallel to the principal planes and opposite corners at (0,0,0) and (2,2,2). Verify the divergence theorem. 5. Verify the Stoke's theorem for the vector field A = & 3xy² + ŷ xy³, where C is the polygonal path from (0,0) to (1,0) to (0,1) to (0,0).
Transcribed Image Text:1. Find the gradient of the following functions at the given point and hence find the directional derivative in the direction of a given vector. a) T(x,y,z) = cosx at (0, –1,3) along d = & – 3ŷ + 2 2. (x²+y²+z²) b) T(x,y,z) = Vx² +z² + In xyz at (1, 2, 1) along d = 3 ŷ +4 2. 2. Find the divergence and curl of the following functions a) A = & tan-1x +ŷ In sin y² + 2 sin z b) A = & cos²(x² +y³) +ŷ In (=) + 2 etan-z. 3. Find the Laplacian of the following functions a) T(x,y,z) = /x2 + y² + z² , b) T(x,y, z) = 3xy²+ y cos z + e. 4. Let A = & x?y – ŷ 3xy + 2 z², and consider the three-dimensional volume inside the cube with faces parallel to the principal planes and opposite corners at (0,0,0) and (2,2,2). Verify the divergence theorem. 5. Verify the Stoke's theorem for the vector field A = & 3xy² + ŷ xy³, where C is the polygonal path from (0,0) to (1,0) to (0,1) to (0,0).
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