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Chapter 6: Q19P (page 295)

As in Problem 17, find the following gradients in two ways and show that your answers are equivalent. $鈭� y$

Short Answer

Expert verified

$鈭� y = \hat{j}$

Step by step solution

01

Given Information

The given information is $鈭� y$ .

02

Definition of Scalar field

In mathematics and physics, scalar field or scalar-valued function referred to a scalarvalueto everypointin aspace鈥� possiblyphysical space. The scalar may either be a (dimensionless)mathematical numberor aphysical quantity.

03

Find the gradients

Use the equation . $鈭� f = e_{r} \frac{鈭� f}{鈭� r} + e_{胃} \frac{1}{r} \frac{鈭� f}{鈭� 胃} + e_{z} \frac{鈭� f}{鈭� z}$

$\begin{array}{rcl} 鈭� y & = & 鈭� r s i n (胃) \\ = & \frac{鈭� r s i n (胃)}{鈭� r} e_{r} + \frac{鈭� r s i n (胃)}{鈭� r} \frac{1}{r} e_{胃} 鈭� y \\ = & s i n (胃) e_{r} + \cos (胃) e_{胃} \end{array}$

Simplify further.

$\begin{array}{rcl} 鈭� y & = & s i n (胃) [\cos (胃) \hat{i} + s i n (胃) \hat{j}] + \cos (胃) [- s i n (胃) \hat{i} + \cos (胃) \hat{j}] \\ = & \hat{j} \end{array}$

Hence, the solution to this problem is . $鈭� y = \hat{j}$

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Most popular questions from this chapter

Evaluate each of the integrals in Problems 3 to 8 as either a volume integral or a surface integral, whichever is easier.

$鈭� r . 苍诲蟽$ Over the whole surface of the cylinder bounded by $x^{2} + y^{2} =, z = 0, and z = 3; r means ix + jy + k z$

Find vector fields $A$ such that role="math" localid="1657346627450" $V = curl A$ for each givenrole="math" localid="1657346639484" $V .$

Evaluate each of the following integrals in the easiest way you can.

$鈭� (y^{2} - x^{2}) d x + (2 x y + 3) d y$ ,along the xaxis from (0,0) and localid="1659182150932" $(鈭� 5, 0)$ then along

a circular arc from $(鈭� 5, 0) to (1, 2) .$

Find vector fields $A$ such that $V = curl A$ for each given V. $V = (y + z) i + (x 鈭� z) j + (x^{2} + y^{2}) k$

Evaluate each of the integrals in Problems $3$ to $8$ as either a volume integral or a surface integral, whichever is easier.

$鈭� (鈭� 脳 F) 诲蟿$ over the region $x^{2} + y^{2} + z^{2} 鈮� 25$ , where localid="1657282505088" $F = (x^{2} + y^{2} + z^{2}) (xi + yj + zk)$

See all solutions

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