Hey i have a calculus test tomorrow and im having trouble understanding scalar fields and vector fields so please try to explain these.
Let f be a scalar field and F a vector field. State whether each expression make sense. If not, explain why. If so, state whether it is a scalar field or a vector field.
a) curl f
b) grad f
c) div F
d) curl(grad f)
e) grad F
f) grad(div F)
g) div(grad f)
h) grad(div f)
i) curl(curl F)
j) div(div F)
k) (grad f) X (div F)
l) div(curl(grad f))
Let f be a scalar field and F a vector field. State whether each expression make sense. If not, explain why. If so, state whether it is a scalar field or a vector field.
a) curl f
b) grad f
c) div F
d) curl(grad f)
e) grad F
f) grad(div F)
g) div(grad f)
h) grad(div f)
i) curl(curl F)
j) div(div F)
k) (grad f) X (div F)
l) div(curl(grad f))
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a) No. curl operates on vectors.
b) Yes. grad f is a vector,
c) Yes. div operates on vectors. div F is a scalar.
d) Yes. See a) and b) above. curl(grad(f)) is a vector.
e) No. grad operates on scalars.
f) Yes. grad(div F) is a vector.
g) Yes. div(grad f) is a scalar.
h) No. div operates on vectors.
i) Yes. curl(curl(F)) is a vector.
j) No. div F is a scalar, and div operates on vectors (and outputs a vector).
k) If by X you mean vector cross product, no. grad f is a vector, but div F is a scalar. The cross product is an operation between two vectors.
l) Yes. div(curl(grad f)) is a scalar.
b) Yes. grad f is a vector,
c) Yes. div operates on vectors. div F is a scalar.
d) Yes. See a) and b) above. curl(grad(f)) is a vector.
e) No. grad operates on scalars.
f) Yes. grad(div F) is a vector.
g) Yes. div(grad f) is a scalar.
h) No. div operates on vectors.
i) Yes. curl(curl(F)) is a vector.
j) No. div F is a scalar, and div operates on vectors (and outputs a vector).
k) If by X you mean vector cross product, no. grad f is a vector, but div F is a scalar. The cross product is an operation between two vectors.
l) Yes. div(curl(grad f)) is a scalar.