Sure, let's go through these vector calculus identities step by step.

1. ∇ × (F + G) = ∇ × F + ∇ × G

This is a direct application of the distributive law of vector calculus. The curl operator (∇ ×) distributes over vector addition just like ordinary multiplication distributes over ordinary addition. So, the curl of the sum of two vectors (F and G) is equal to the sum of the curl of each vector.

2. ∇ • (F × G) = G • (∇ × F) − F • (∇ × G)

This is known as the vector triple product rule (or the BAC-CAB rule). It states that the divergence of the cross product of two vectors (F and G) is equal to the dot product of the second vector and the curl of the first vector minus the dot product of the first vector and the curl of the second vector.

This can be proven by writing out the components of the vectors and the operators in Cartesian coordinates and applying the definitions of the dot product, cross product, and curl. The details of this calculation are a bit lengthy, but it's a standard result in vector calculus.

Question

Sure, let's go through these vector calculus identities step by step.

1. ∇ × (F + G) = ∇ × F + ∇ × G

This is a direct application of the distributive law of vector calculus. The curl operator (∇ ×) distributes over vector addition just like ordinary multiplication distributes over ordinary addition. So, the curl of the sum of two vectors (F and G) is equal to the sum of the curl of each vector.

2. ∇ • (F × G) = G • (∇ × F) − F • (∇ × G)

This is known as the vector triple product rule (or the BAC-CAB rule). It states that the divergence of the cross product of two vectors (F and G) is equal to the dot product of the second vector and the curl of the first vector minus the dot product of the first vector and the curl of the second vector.

This can be proven by writing out the components of the vectors and the operators in Cartesian coordinates and applying the definitions of the dot product, cross product, and curl. The details of this calculation are a bit lengthy, but it's a standard result in vector calculus.

Knowee AI · Accepted Answer

Sure, let's go through these vector calculus identities step by step.

1. ∇ × (F + G) = ∇ × F + ∇ × G

This is a direct application of the distributive law of vector calculus. The curl operator (∇ ×) distributes over vector addition just like ordinary multiplication distributes over ordinary addition. So, the curl of the sum of two vectors (F and G) is equal to the sum of the curl of each vector.

2. ∇ • (F × G) = G • (∇ × F) − F • (∇ × G)

This is known as the vector triple product rule (or the BAC-CAB rule). It states that the divergence of the cross product of two vectors (F and G) is equal to the dot product of the second vector and the curl of the first vector minus the dot product of the first vector and the curl of the second vector.

This can be proven by writing out the components of the vectors and the operators in Cartesian coordinates and applying the definitions of the dot product, cross product, and curl. The details of this calculation are a bit lengthy, but it's a standard result in vector calculus.

For general vector ﬁelds F and G prove that∇ × (F + G) = ∇ × F + ∇ × G,[6 marks]∇ • (F × G) = G • (∇ × F) − F • (∇ × G)

Question

For general vector ﬁelds F and G prove that $\nabla \times (F + G) = \nabla \times F + \nabla \times G$ , [6 marks] ( \nabla \cdot (F \times G) = G \cdot (\nabla \times F) - F \cdot (\nabla \times G) \

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