Given scalar field:T(x,y,z) = 2-xyz3Solve Grad(T) at point P ( 8 , 2.1 rad , 2.3 rad ) Please do the following depending on the final answer:Scalar: Write the answer as is (include negative symbol if the number is negative)Vector: Get the magnitude of the vectorInvalid: Write the number "0.005" as your answer in the field provided.Variables
Question
Given scalar field:T(x,y,z) = 2-xyz3Solve Grad(T) at point P ( 8 , 2.1 rad , 2.3 rad ) Please do the following depending on the final answer:Scalar: Write the answer as is (include negative symbol if the number is negative)Vector: Get the magnitude of the vectorInvalid: Write the number "0.005" as your answer in the field provided.Variables
Solution
The gradient of a scalar field T(x, y, z) is a vector field that points in the direction of the greatest rate of increase of T, and whose magnitude is the rate of change in that direction.
The gradient of T(x, y, z) = 2 - xyz^3 is given by the vector (∂T/∂x, ∂T/∂y, ∂T/∂z).
We can compute the partial derivatives as follows:
∂T/∂x = -yz^3 ∂T/∂y = -xz^3 ∂T/∂z = -3xyz^2
Substituting the point P(8, 2.1 rad, 2.3 rad) into these expressions, we get:
∂T/∂x = -(2.1 rad)(2.3 rad)^3 = -11.0877 rad ∂T/∂y = -(8)(2.3 rad)^3 = -85.184 rad ∂T/∂z = -3(8)(2.1 rad)(2.3 rad)^2 = -285.408 rad
So, Grad(T) at point P is the vector (-11.0877 rad, -85.184 rad, -285.408 rad).
The magnitude of this vector is given by sqrt((∂T/∂x)^2 + (∂T/∂y)^2 + (∂T/∂z)^2) = sqrt((-11.0877 rad)^2 + (-85.184 rad)^2 + (-285.408 rad)^2) = 297.97 rad.
Similar Questions
Given vector field:R(x,y,z) = (xy3z2 ) ax + ( 2-xyz3) ay +(xyz) azSolve Grad(Div(R)) at point P ( 5 , -4 , 5 ) Please do the following depending on the final answer:Scalar: Write the answer as is (include negative symbol if the number is negative)Vector: Get the magnitude of the vectorInvalid: Write the number "0.005" as your answer in the field provided.
Given vector field:R(x,y,z) = (xy3z2 ) ax + ( 2-xyz3) ay +(xyz) azSolve (Curl(R)) at point P ( -1 , -2 , 1 ) Please do the following depending on the final answer:Scalar: Write the answer as is (include negative symbol if the number is negative)Vector: Get the magnitude of the vectorInvalid: Write the number "0.005" as your answer in the field provided.
Given the vector u defined by the points:(−7,4),(−2,−6)
Write an equation of the plane with normal vector n =⟨−7,−4,−6⟩ passing through the point (−8,4,6) in scalar form answer box =20
Given vector R = 3 ax - 4 ay - 2az at point P(4,5 , 2)Determine Rɸ
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.