Kate asked in Science & MathematicsMathematics · 8 years ago

# gradient question for calculus 3?

Let R = x i + y j + z k and r = |R|. Find each of the following.

(a) ∇r

[options]

1. -R/r^3

2. R/r

3. R/r^2

4. r/R

5. 0

(b) ∇(1/r)

[options]

1. -R/r^3

2. R/r^2

3. R/r

4. r/R

5. 0

(c) ∇ln r

[options]

1. 0

2. R/r^3

3. R/r

4. R/r^2

5. r/R

thanks if you can help me with any of them!!

Relevance
• 8 years ago

(a)

R = x i + y j + z k

∂R/∂x = i, ∂R/∂y = j, ∂R/∂z = k

∇R = i+j+k

Now apply chain rule.

∇r = ∇|R| = ∂|R|/∂R · ∇R = R/|R| = R/r

(b)

Similarly, apply chain rule.

∇(1/r) = ∇(r⁻¹) = - 1/r² * ∇r = - 1/r² * R/r = - R/r³

(c)

Again apply chain rule.

∇(lnr) = 1/r * ∇r = 1/r * R/r = R/r²

• kb
Lv 7
8 years ago

Note that r = ||R|| = √(x^2 + y^2 + z^2).

1) ∇r = <∂r/∂x, ∂r/∂y, ∂r/∂z>

........= <x/√(x^2 + y^2 + z^2), y/√(x^2 + y^2 + z^2), z/√(x^2 + y^2 + z^2)>

........= R/r.

2) ∇(1/r)

.....= ∇((x^2 + y^2 + z^2)^(-1/2))

.....= <-x/(x^2 + y^2 + z^2)^(3/2), -y/(x^2 + y^2 + z^2)^(3/2), -z/(x^2 + y^2 + z^2)^(3/2)>

.....= -R/r^3.

2) ∇(ln r)

.....= ∇((1/2) ln(x^2 + y^2 + z^2))

.....= <x/(x^2 + y^2 + z^2), y/(x^2 + y^2 + z^2), z/(x^2 + y^2 + z^2)>

.....= R/r^2.

I hope this helps!