Chapter 6: Q1P (page 294)

Find the gradient of $w = x^{2} y^{3} z$ at $(1, 2, - 1)$ .

Short Answer

Expert verified

The gradient of function $w = x^{2} y^{3} z$ at $(1, 2, - 1)$ is $- 16 i - 12 j + 8 k$ .

Step by step solution

Given Information.

The given function is $w = x^{2} y^{3} z$ and reference point is $(1, 2, - 1)$ .

Definition of Gradient.

Gradient is a differential operator which is applied to vector function in three dimensions in order to give a vector whose three components are the partial derivatives of given function with respect to its three variables. It is represented by $\nabla$ .

Use the formula.

The formula states the expression mentioned below.

$\begin{array}{l} \nabla w = g r a d w \\ \nabla w = i \frac{\partial w}{\partial x} + y \frac{\partial w}{\partial y} + k \frac{\partial w}{\partial z} \end{array}$

Calculate the gradient.

Use the gradient formula for the given vector $w = x^{2} y^{3} z$ .

$\begin{array}{l} \nabla w = i \frac{\partial}{\partial x} (x^{2} y^{3} z) + j \frac{\partial}{\partial y} (x^{2} y^{3} z) + k \frac{\partial}{\partial z} (x^{2} y^{3} z) \\ \nabla w = i (2 x y^{3} z) + j (3 x^{2} y^{2} z) + k (x^{2} y^{3}) \end{array}$

Use the reference point given below.

$\begin{array}{l} x = 1, \\ y = 2, \\ z = - 1 \end{array}$

Now put the values in the equation.

$\begin{array}{l} \nabla w_{(1, 2, - 1)} = 2 (1) {(2)}^{3} (- 1) i + 3 {(1)}^{2} {(2)}^{2} (- 1) j + {(1)}^{2} {(2)}^{3} k \\ \nabla w_{(1, 2, - 1)} = - 16 i - 12 j + 8 k \end{array}$

The gradient of function $w = x^{2} y^{3} z$ at $(1, 2, - 1)$ is $- 16 i - 12 j + 8 k$ .

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Find the gradient of $w = x^{2} y^{3} z$ at $(1, 2, - 1)$ .

Short Answer

Step by step solution

Given Information.

Definition of Gradient.

Use the formula.

Calculate the gradient.

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Find the gradient of w=x2y3z at (1,2,-1).

Short Answer

Step by step solution

Given Information.

Definition of Gradient.

Use the formula.

Calculate the gradient.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Find the gradient of $w = x^{2} y^{3} z$ at $(1, 2, - 1)$ .