Given the scalar function (x, y, z) = e^xy - x2yz2.

Given the scalar function (x, y, z) = e^xy - x^2yz^2. The gradient of a scalar function (x, y, z) is given by: = ( )/( x) i + ( )/( y) j + ( )/( z) k a) Evaluate (x, y, z) Step 1: Calculate the partial derivative with respect to x. ( )/( x) = ()/( x) (e^xy - x^2yz^2) = y e^xy - 2xyz^2 Step 2: Calculate the partial derivative with respect to y. ( )/( y) = ()/( y) (e^xy - x^2yz^2) = x e^xy - x^2z^2 Step 3: Calculate the partial derivative with respect to z. ( )/( z) = ()/( z) (e^xy - x^2yz^2) = -x^2y(2z) = -2x^2yz Step 4: Combine the partial derivatives to form the gradient vector. (x, y, z) = (y e^xy - 2xyz^2) i + (x e^xy - x^2z^2) j + (-2x^2yz) k The gradient is: (x, y, z) = (y e^xy - 2xyz^2) i + (x e^xy - x^2z^2) j - 2x^2yz k b) Evaluate (2, -1, 1) Substitute x=2, y=-1, and z=1 into the expression for (x, y, z) from part (a). Step 1: Calculate the i component. y e^xy - 2xyz^2 = (-1) e^(2)(-1) - 2(2)(-1)(1)^2 = -e^-2 + 4 Step 2: Calculate the j component. x e^xy - x^2z^2 = (2) e^(2)(-1) - (2)^2(1)^2 = 2e^-2 - 4 Step 3: Calculate the k component. -2x^2yz = -2(2)^2(-1)(1) = -2(4)(-1)(1) = 8 Step 4: Combine the components. (2, -1, 1) = (-e^-2 + 4) i + (2e^-2 - 4) j + 8 k The gradient at (2, -1, 1) is: (2, -1, 1) = (4 - e^-2) i + (2e^-2 - 4) j + 8 k c) Evaluate | (1, 1, -1)| First, evaluate (1, 1, -1) by substituting x=1, y=1, and z=-1 into the expression for (x, y, z). Step 1: Calculate the i component. y e^xy - 2xyz^2 = (1) e^(1)(1) - 2(1)(1)(-1)^2 = e - 2(1)(1) = e - 2 Step 2: Calculate the j component. x e^xy - x^2z^2 = (1) e^(1)(1) - (1)^2(-1)^2 = e - 1(1) = e - 1 Step 3: Calculate the k component. -2x^2yz = -2(1)^2(1)(-1) = -2(1)(-1) = 2 Step 4: Form the gradient vector at (1, 1, -1). (1, 1, -1) = (e - 2) i + (e - 1) j + 2 k Step 5: Calculate the magnitude of the vector. The magnitude of a vector Ai + Bj + Ck is sqrt(A^2 + B^2 + C^2). | (1, 1, -1)| = sqrt((e-2)^2 + (e-1)^2 + (2)^2) | (1, 1, -1)| = sqrt((e^2 - 4e + 4) + (e^2 - 2e + 1) + 4) | (1, 1, -1)| = sqrt(e^2 - 4e + 4 + e^2 - 2e + 1 + 4) | (1, 1, -1)| = sqrt(2e^2 - 6e + 9) The magnitude of the gradient at (1, 1, -1) is: | (1, 1, -1)| = sqrt(2e^2 - 6e + 9)