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!!

2 Answers

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  • 8 years ago
    Favorite Answer

    (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!

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