(3,-4,-2) Show that 323 f dx + gdy + hdz = f(-3) 12x²y^z²dx + 16y³x³z²dy + 8zy r³dz is independent of path: (-4,2,-3) (-4,2,-3) || || || || || || Therefore curl F = (2-3) 12x²y²z²dx + 16y³x³z²dy + 8zy^¹r³dz = (
(3,-4,-2) Show that 323 f dx + gdy + hdz = f(-3) 12x²y^z²dx + 16y³x³z²dy + 8zy r³dz is independent of path: (-4,2,-3) (-4,2,-3) || || || || || || Therefore curl F = (2-3) 12x²y²z²dx + 16y³x³z²dy + 8zy^¹r³dz = (
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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![The image contains a mathematical exercise focusing on vector calculus and integration. Below is the transcription of the content in the image:
---
**Show that:**
\[
\int_{(-4, 2, -3)}^{(3, -4, -2)} f \, dx + g \, dy + h \, dz = \int_{(-4, 2, -3)}^{(3, -4, -2)} 12x^2y^4z^2 \, dx + 16y^3x^3z^2 \, dy + 8zy^4x^3 \, dz
\]
**is independent of path:**
\[
\frac{\partial h}{\partial y} = \boxed{\phantom{aaa}}
\]
\[
\frac{\partial g}{\partial z} = \boxed{\phantom{aaa}}
\]
\[
\frac{\partial f}{\partial z} = \boxed{\phantom{aaa}}
\]
\[
\frac{\partial h}{\partial z} = \boxed{\phantom{aaa}}
\]
\[
\frac{\partial g}{\partial x} = \boxed{\phantom{aaa}}
\]
\[
\frac{\partial f}{\partial y} = \boxed{\phantom{aaa}}
\]
**Therefore curl F =**
\[
\boxed{\phantom{aaaaaaaa}}
\]
\[
\int_{(-4, 2, -3)}^{(3, -4, -2)} 12x^2y^4z^2 \, dx + 16y^3x^3z^2 \, dy + 8zy^4x^3 \, dz = \boxed{\phantom{aaaaaaaa}}
\]
---
The task involves determining whether a given line integral is path-independent by checking the conditions for the curl of the vector field. The exercise aims to demonstrate this concept using partial derivatives and integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e0c8037-834b-4d36-b3c4-eda43340784b%2F68ac6849-e0ee-46a0-b8d5-23f2d413abc7%2Fmzwomcl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical exercise focusing on vector calculus and integration. Below is the transcription of the content in the image:
---
**Show that:**
\[
\int_{(-4, 2, -3)}^{(3, -4, -2)} f \, dx + g \, dy + h \, dz = \int_{(-4, 2, -3)}^{(3, -4, -2)} 12x^2y^4z^2 \, dx + 16y^3x^3z^2 \, dy + 8zy^4x^3 \, dz
\]
**is independent of path:**
\[
\frac{\partial h}{\partial y} = \boxed{\phantom{aaa}}
\]
\[
\frac{\partial g}{\partial z} = \boxed{\phantom{aaa}}
\]
\[
\frac{\partial f}{\partial z} = \boxed{\phantom{aaa}}
\]
\[
\frac{\partial h}{\partial z} = \boxed{\phantom{aaa}}
\]
\[
\frac{\partial g}{\partial x} = \boxed{\phantom{aaa}}
\]
\[
\frac{\partial f}{\partial y} = \boxed{\phantom{aaa}}
\]
**Therefore curl F =**
\[
\boxed{\phantom{aaaaaaaa}}
\]
\[
\int_{(-4, 2, -3)}^{(3, -4, -2)} 12x^2y^4z^2 \, dx + 16y^3x^3z^2 \, dy + 8zy^4x^3 \, dz = \boxed{\phantom{aaaaaaaa}}
\]
---
The task involves determining whether a given line integral is path-independent by checking the conditions for the curl of the vector field. The exercise aims to demonstrate this concept using partial derivatives and integration.
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