True or False (i) Let f : R² —>R be a function. If both df/dx and df/dy exist at ā∈R², then any directional derivative of f also exists at ā (ii) Let D be a closed and bounded region in R² and let f be a C¹ function over D. Suppose that f attains its minimum over D at a point ā ∈ D, then ▽f (ā) = 0. (iii) Let A be a subset of R². If A is not open, then A is closed. (iv) Let F be a C² vector field on R³. Then div(curl F) must be the zero vector field. (v) Let f be a C² function on Rⁿ and let ā be a critical point of f. If Hess f(ā) has both positive and negative eigenvalues, then ā cannot be a local minimum or maximum of f.
True or False (i) Let f : R² —>R be a function. If both df/dx and df/dy exist at ā∈R², then any directional derivative of f also exists at ā (ii) Let D be a closed and bounded region in R² and let f be a C¹ function over D. Suppose that f attains its minimum over D at a point ā ∈ D, then ▽f (ā) = 0. (iii) Let A be a subset of R². If A is not open, then A is closed. (iv) Let F be a C² vector field on R³. Then div(curl F) must be the zero vector field. (v) Let f be a C² function on Rⁿ and let ā be a critical point of f. If Hess f(ā) has both positive and negative eigenvalues, then ā cannot be a local minimum or maximum of f.
True or False (i) Let f : R² —>R be a function. If both df/dx and df/dy exist at ā∈R², then any directional derivative of f also exists at ā (ii) Let D be a closed and bounded region in R² and let f be a C¹ function over D. Suppose that f attains its minimum over D at a point ā ∈ D, then ▽f (ā) = 0. (iii) Let A be a subset of R². If A is not open, then A is closed. (iv) Let F be a C² vector field on R³. Then div(curl F) must be the zero vector field. (v) Let f be a C² function on Rⁿ and let ā be a critical point of f. If Hess f(ā) has both positive and negative eigenvalues, then ā cannot be a local minimum or maximum of f.
True or False
(i) Let f : R² —>R be a function. If both df/dx and df/dy exist at ā∈R², then any directional derivative of f also exists at ā
(ii) Let D be a closed and bounded region in R² and let f be a C¹ function over D. Suppose that f attains its minimum over D at a point ā ∈ D, then ▽f (ā) = 0.
(iii) Let A be a subset of R². If A is not open, then A is closed.
(iv) Let F be a C² vector field on R³. Then div(curl F) must be the zero vector field.
(v) Let f be a C² function on Rⁿ and let ā be a critical point of f. If Hess f(ā) has both positive and negative eigenvalues, then ā cannot be a local minimum or maximum of f.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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