Determine if F is conservative, if it is then find a potential function. (a) F =< 2x + ye", 2y + yeª > (b) F =< y,x, z² > (c) F =< cos z, 2y, –x sin z >

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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### Determining if a Vector Field is Conservative and Finding a Potential Function

#### Problem Statement
Determine if the vector field **F** is conservative. If it is, then find a potential function.

#### Vector Fields to Analyze

(a) **F** = ⟨2x + ye^x, 2y + ye^x⟩

(b) **F** = ⟨y, x, z²⟩

(c) **F** = ⟨cos z, 2y, −x sin z⟩

### Explanation

To determine if a vector field **F** is conservative, check if the curl of **F** is zero. If the vector field is conservative, then there exists a potential function φ such that **F** = ∇φ. The task involves verifying the conservativeness and finding the corresponding potential function for each of the given vector fields.
Transcribed Image Text:### Determining if a Vector Field is Conservative and Finding a Potential Function #### Problem Statement Determine if the vector field **F** is conservative. If it is, then find a potential function. #### Vector Fields to Analyze (a) **F** = ⟨2x + ye^x, 2y + ye^x⟩ (b) **F** = ⟨y, x, z²⟩ (c) **F** = ⟨cos z, 2y, −x sin z⟩ ### Explanation To determine if a vector field **F** is conservative, check if the curl of **F** is zero. If the vector field is conservative, then there exists a potential function φ such that **F** = ∇φ. The task involves verifying the conservativeness and finding the corresponding potential function for each of the given vector fields.
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