Problem 1.4 Using the following vectors A = a i+ 2y j+ 3z k B = 3y i – 2x j Check the product rules by calculating each term separately a) Divergence Product Rule (iv) V· (A x B) = B· (V × A) – A · (V × B) %3D
Problem 1.4 Using the following vectors A = a i+ 2y j+ 3z k B = 3y i – 2x j Check the product rules by calculating each term separately a) Divergence Product Rule (iv) V· (A x B) = B· (V × A) – A · (V × B) %3D
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![## Problem 1.4
**Using the following vectors:**
\[
\mathbf{A} = x \, \mathbf{i} + 2y \, \mathbf{j} + 3z \, \mathbf{k}
\]
\[
\mathbf{B} = 3y \, \mathbf{i} - 2x \, \mathbf{j}
\]
Check the product rules by calculating each term separately:
### a) Divergence Product Rule (iv)
\[
\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B})
\]
### b) Gradient Product Rule (ii)
\[
\nabla (\mathbf{A} \cdot \mathbf{B}) = \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) + (\mathbf{A} \cdot \nabla) \mathbf{B} + (\mathbf{B} \cdot \nabla) \mathbf{A}
\]
### c) Curl Product Rule (vi)
\[
\nabla \times (\mathbf{A} \times \mathbf{B}) = (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} + \mathbf{A} (\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \mathbf{A})
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72a720b9-38b2-4beb-ae1b-56a99a6fccf3%2Fadcbe656-1e37-499d-828b-54de23e5bf9e%2Fqrkmku9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Problem 1.4
**Using the following vectors:**
\[
\mathbf{A} = x \, \mathbf{i} + 2y \, \mathbf{j} + 3z \, \mathbf{k}
\]
\[
\mathbf{B} = 3y \, \mathbf{i} - 2x \, \mathbf{j}
\]
Check the product rules by calculating each term separately:
### a) Divergence Product Rule (iv)
\[
\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B})
\]
### b) Gradient Product Rule (ii)
\[
\nabla (\mathbf{A} \cdot \mathbf{B}) = \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) + (\mathbf{A} \cdot \nabla) \mathbf{B} + (\mathbf{B} \cdot \nabla) \mathbf{A}
\]
### c) Curl Product Rule (vi)
\[
\nabla \times (\mathbf{A} \times \mathbf{B}) = (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} + \mathbf{A} (\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \mathbf{A})
\]
![**Problem 1.5**
Using the formal definition of the Laplacian,
\[
\nabla^2 T = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2}
\]
Calculate the Laplacian of the following functions:
a) \( T_a = x^2 + 2xy + 3z + 4 \)
b) \( T_b = \sin x \sin y \sin z \)
c) \( T_c = e^{-5x} \sin 4y \cos 3z \)
d) \( \mathbf{v} = x^2 \, \mathbf{i} + 3xz^2 \, \mathbf{j} - 2xz \, \mathbf{k} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72a720b9-38b2-4beb-ae1b-56a99a6fccf3%2Fadcbe656-1e37-499d-828b-54de23e5bf9e%2Fxb74ftw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1.5**
Using the formal definition of the Laplacian,
\[
\nabla^2 T = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2}
\]
Calculate the Laplacian of the following functions:
a) \( T_a = x^2 + 2xy + 3z + 4 \)
b) \( T_b = \sin x \sin y \sin z \)
c) \( T_c = e^{-5x} \sin 4y \cos 3z \)
d) \( \mathbf{v} = x^2 \, \mathbf{i} + 3xz^2 \, \mathbf{j} - 2xz \, \mathbf{k} \)
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